- Each expression contains like terms. - Combine the numerical coefficients. - Keep track of each coefficient’s sign.

1. \(12-7+3=8\), so the result is \(8a\). 2. \(-5+9-6=-2\), so the result is \(-2x^2\). 3. \(4.5-1.2-2.1=1.2\), so the result is \(1.2y\).

1) \(8a\) 2) \(-2x^2\) 3) \(1.2y\)

- Confirm that every term has the same variables and exponents. - Rewrite additions of negative terms as subtraction. - Combine the signed coefficients.

1. All terms have the same variable part \(ab^2\). 2. Combine the coefficients: \(12-5-8-2=-3\). 3. The simplified expression is \(-3ab^2\).

\(-3ab^2\)

- Subtracting a negative term changes to addition. - Combine the signed coefficients. - Opposite terms cancel.

1. \(12x^2-(-8x^2)=12x^2+8x^2=20x^2\). 2. \(-15y^3-(-10y^3)=-15y^3+10y^3=-5y^3\). 3. \(7a^4-(+11a^4)=7a^4-11a^4=-4a^4\). 4. \(-b^2-(-b^2)=-b^2+b^2=0\).

1) \(20x^2\) 2) \(-5y^3\) 3) \(-4a^4\) 4) \(0\)

- Rewrite each repeated product as a power. - Combine only terms with identical variable parts. - Count how many copies of each like term appear.

1. \(k^2 + k^2 + k^2 = 3k^2\). 2. \(x^3 + y^2 + x^3 + y^2 = 2x^3 + 2y^2\). 3. \(a^2 + a^2 + a^2 = 3a^2\). 4. \(m^3 + n^2 + m^3 = 2m^3 + n^2\).

1) \(3k^2\) 2) \(2x^3 + 2y^2\) 3) \(3a^2\) 4) \(2m^3 + n^2\)

- Compare the variable part of each term, not just its coefficient. - Like terms must have matching exponents.

1. Like terms must have the same variable part with the same exponent. 2. The terms \(4x^2\) and \(5x^2\) both have variable part \(x^2\), so they are like terms.

\(4x^2\) and \(5x^2\)

- When subtracting parentheses, change the sign of every term inside. - Identify like terms by matching both their variables and exponents. - Expand all products before combining terms. - A negative factor times a negative term gives a positive term.

1. For a), \(15x - 4y + 10 - 6x - 2y + 3 = 9x - 6y + 13\). 2. For b), \(4.2a^2 - 2a + 1.8a^2 + 5a = 6a^2 + 3a\). 3. For c), \(12 - 3x + 12 + 10 - 4x = 34 - 7x\).

a) \(9x - 6y + 13\) b) \(6a^2 + 3a\) c) \(34 - 7x\)

- Compare both the variables and their exponents when identifying like terms. - Rewrite the fractional coefficients with a common denominator. - Combine each group of like terms separately.

1. The \(x^2y\)-terms and \(xy^2\)-terms are not like terms, so combine them separately. 2. Combine the \(x^2y\)-terms: \(2.4x^2y+0.6x^2y=3x^2y\). 3. Combine the \(xy^2\)-terms: \(-\frac{3}{5}+\frac{1}{10}=-\frac{6}{10}+\frac{1}{10}=-\frac{5}{10}=-0.5\). 4. The simplified expression is \(3x^2y-0.5xy^2\).

\(3x^2y-0.5xy^2\)

- An \(a^3\)-term with no visible coefficient has coefficient \(1\). - Convert fractions and decimals to a common form. - Combine terms only when their variable parts and exponents match.

1. Combine the \(a^3\)-terms: \(\frac{2}{3}+\frac{1}{6}-1=\frac{4}{6}+\frac{1}{6}-\frac{6}{6}=-\frac{1}{6}\). 2. Combine the \(a^2\)-terms: \(-0.75+\frac{1}{4}=-0.75+0.25=-0.5\). 3. The simplified polynomial is \(-\frac{1}{6}a^3-0.5a^2\).

\(-\frac{1}{6}a^3-0.5a^2\)

- Like terms have the same variables raised to the same powers. - Combine the coefficients while keeping their signs. - The constant term remains separate.

1. Combine the \(x^2\)-terms: \(0.7x^2+1.3x^2=2x^2\). 2. Combine the \(x\)-terms: \(-5x+2x=-3x\). 3. Keep the constant term \(-11\). The simplified polynomial is \(2x^2-3x-11\).

\(2x^2-3x-11\)

- Each part contains like terms with the same variable and exponent. - Convert mixed numbers to decimals or improper fractions. - A term disappears when its coefficient is zero.

1. For part a, convert \(1\frac{4}{5}\) to \(1.8\). Then \(4.2-6+1.8=0\), so the expression equals \(0\). 2. For part b, convert \(2\frac{3}{4}\) to \(2.75\). Then \(2.75-1.5-3=-1.75\), so the result is \(-1.75x^2\).

a) \(0\) b) \(-1.75x^2\), or \(-1\frac{3}{4}x^2\)

- Use a common denominator to combine fractional coefficients. - Keep track of every negative sign. - Simplify the final coefficient when possible.

1. For part a, use denominator \(12\): \(\frac{7}{12}-\frac{3}{12}-\frac{8}{12}=-\frac{4}{12}=-\frac{1}{3}\). The result is \(-\frac{1}{3}a^3\). 2. For part b, use denominator \(8\): \(-\frac{4}{8}+\frac{5}{8}-\frac{6}{8}=-\frac{5}{8}\). The result is \(-\frac{5}{8}b^4\).

a) \(-\frac{1}{3}a^3\) b) \(-\frac{5}{8}b^4\)

- Each group consists of like terms. - Combine only the coefficients. - Remember that coefficients of \(1\), \(-1\), and \(0\) have special written forms.

1. \(4.2-5.8+0.6=-1\), so the result is \(-x^2\). 2. \(-0.75+1.5-0.25=0.5\), so the result is \(0.5ab^2\). 3. \(12-15-3+7=1\), so the result is \(y^5\). 4. \(0.15-0.4+0.25=0\), so the result is \(0\).

1) \(-x^2\) 2) \(0.5ab^2\) 3) \(y^5\) 4) \(0\)

- Identify which terms are like terms before comparing coefficients. - Treat each missing value as an unknown that must make the whole statement true. - Substitute your result into the original expression to check it.

1. Let the missing coefficient be \(k\). The known coefficients add to \(2.5+1.2=3.7\), so \(3.7-k=0.7\). Thus \(k=3\), and the missing term is \(3a^3b\). 2. Let the missing coefficient be \(c\). Combine the known coefficients: \(3.4-5=-1.6\). Then \(c-1.6=-2.1\), so \(c=-0.5\).

1) \(3a^3b\) 2) \(-0.5\)

- Like terms must have identical variable parts and exponents. - Use the commutative property to recognize that \(ab=ba\). - Combine constant terms separately.

1. Combine the \(m\)-terms and \(n\)-terms: \(10m-4m=6m\) and \(-3n+7n=4n\). The result is \(6m+4n\). 2. Combine terms with matching powers: \(2x^2-7x^2=-5x^2\) and \(5x-x=4x\). The result is \(-5x^2+4x\). 3. Since \(ba=ab\), \(12ab-5ba=7ab\). The term \(4a\) is not like an \(ab\)-term, so the result is \(7ab+4a\). 4. Combine the \(z\)-terms and constants: \(\frac{3}{8}z-\frac{1}{8}z=\frac{1}{4}z\) and \(\frac{1}{4}+\frac{3}{4}=1\). The result is \(\frac{1}{4}z+1\).

1) \(6m+4n\) 2) \(-5x^2+4x\) 3) \(7ab+4a\) 4) \(\frac{1}{4}z+1\)

- Group terms with the same variable parts and exponents. - Keep each term’s sign with its coefficient. - Terms with different powers cannot be combined.

1. \(18x^2-5x^2=13x^2\), and \(7x-7x=0\). The result is \(13x^2\). 2. \(-4a-3a=-7a\), and \(-9+2=-7\). The result is \(-7a-7\). 3. \(5y^3-3y^3=2y^3\), and \(-2y+4y=2y\). The result is \(2y^3+2y\). 4. \(-2m^2+5m^2=3m^2\), and \(-n-4n=-5n\). The result is \(3m^2-5n\).

1) \(13x^2\) 2) \(-7a-7\) 3) \(2y^3+2y\) 4) \(3m^2-5n\)

- Combine coefficients only when the variable parts match exactly. - Use the commutative property to compare \(pq\) and \(qp\). - Terms can cancel when their coefficients are opposites.

1. Combine by power: \(3.5z^2+1.5z^2=5z^2\) and \(-1.2z-0.8z=-2z\). The result is \(5z^2-2z\). 2. \(\frac{2}{5}b+\frac{3}{5}b=b\), and \(-\frac{1}{2}c-\frac{3}{2}c=-2c\). The result is \(b-2c\). 3. Since \(qp=pq\), \(9pq+2qp=11pq\), and \(-4p-p=-5p\). The result is \(11pq-5p\). 4. \(x^4+2x^4=3x^4\), and the \(x^2\)-terms cancel. The result is \(3x^4\).

1) \(5z^2-2z\) 2) \(b-2c\) 3) \(11pq-5p\) 4) \(3x^4\)

- Like terms have exactly the same variables and exponents. - A variable term with no visible coefficient has coefficient \(1\). - Combine each group of coefficients separately.

1. Combine the \(a\)-terms and \(b\)-terms: \(9a+3a+a=13a\) and \(-4b-7b=-11b\). The result is \(13a-11b\). 2. Combine by power: \(1.2x^2+0.8x^2-x^2=x^2\), and \(-0.5x+1.5x=x\). The result is \(x^2+x\). 3. Combine each variable part separately: \(\frac{3}{8}u^2v+\frac{5}{8}u^2v=u^2v\), and \(-\frac{1}{2}uv^2+\frac{3}{4}uv^2=\frac{1}{4}uv^2\). The result is \(u^2v+\frac{1}{4}uv^2\).

1) \(13a-11b\) 2) \(x^2+x\) 3) \(u^2v+\frac{1}{4}uv^2\)

- Compare the complete variable parts, including exponents. - A group of like terms cancels when its coefficients add to zero. - Keep constant terms separate.

1. Combine the \(x^2y\)-terms and \(xy\)-terms: \(17+3=20\) and \(-8-2+1=-9\). The result is \(20x^2y-9xy\). 2. The \(a^3b\)-coefficients add to \(2.5+0.5-3=0\), and the \(ab^3\)-coefficients add to \(-1.4+1.4=0\). The expression simplifies to \(0\). 3. The \(z^2\)-coefficients add to \(\frac{2}{3}+\frac{1}{3}=1\), and the \(z\)-coefficients add to \(-\frac{1}{4}-\frac{3}{4}=-1\). The result is \(z^2-z+5\).

1) \(20x^2y-9xy\) 2) \(0\) 3) \(z^2-z+5\)

- Group terms with identical variables and exponents. - Use a common denominator for fractional coefficients. - Keep each coefficient’s sign.

1. Group the \(x^2y\)-terms and \(xy^2\)-terms separately. 2. For \(x^2y\), \(\frac{1}{3}+\frac{5}{6}-\frac{1}{6}=\frac{2}{6}+\frac{5}{6}-\frac{1}{6}=1\). 3. For \(xy^2\), \(-\frac{5}{2}-\frac{1}{4}=-\frac{10}{4}-\frac{1}{4}=-\frac{11}{4}\). 4. The simplified expression is \(x^2y-\frac{11}{4}xy^2\).

\(x^2y-\frac{11}{4}xy^2\)

- \(a^2\) and \(ab\) are different variable parts. - The coefficient of \(-a^2\) is \(-1\). - Combine each group of coefficients separately.

1. Combine the \(a^2\)-terms: \(2.4+0.6-1=2\). 2. Combine the \(ab\)-terms: \(-1.5+3=1.5\). 3. The simplified expression is \(2a^2+1.5ab\).

\(2a^2+1.5ab\)

- Combine the signed coefficients of the like terms. - Rewrite fractions with a common denominator. - A term with coefficient zero disappears.

1. \(14-8-6=0\), so the expression equals \(0\). 2. \(-4.2-1.5+0.7=-5\), so the result is \(-5xy\). 3. \(\frac{3}{4}-\frac{1}{2}+\frac{1}{4}=\frac{3}{4}-\frac{2}{4}+\frac{1}{4}=\frac{1}{2}\), so the result is \(\frac{1}{2}b^2\).

1) \(0\) 2) \(-5xy\) 3) \(\frac{1}{2}b^2\)

- Group terms by their powers of \(x\). - Keep each term’s sign. - Like terms must have identical variable parts and exponents.

1. Combine the quadratic terms: \(5x^2-8x^2=-3x^2\). 2. Combine the linear terms: \(-12x+4x=-8x\). 3. Combine the constants: \(9-15=-6\). 4. In descending powers, the result is \(-3x^2-8x-6\). 5. The variable parts \(x^2\) and \(x\) have different exponents, so they are not like terms and cannot be combined by adding their coefficients.

a) \(-3x^2-8x-6\) b) \(x^2\)-terms and \(x\)-terms have different exponents, so they are not like terms.

- Add fractions by using a common denominator. - A variable term with no visible coefficient has coefficient \(1\). - Adding like terms does not change their exponents.

1. Statement 1 is correct because \(5-8=-3\). 2. Statement 2 is incorrect. Use a common denominator: \(\frac{1}{2}x+\frac{1}{4}x=\frac{2}{4}x+\frac{1}{4}x=\frac{3}{4}x\). The original work incorrectly added the denominators. 3. Statement 3 is correct because \(-0.3+1=0.7\). 4. Statement 4 is incorrect. Adding like terms changes the coefficient, not the exponent: \(b^2+b^2=2b^2\). 5. Statement 5 is correct because \(4-1=3\).

1) Correct. 2) Incorrect; \(\frac{3}{4}x\). 3) Correct. 4) Incorrect; \(2b^2\). 5) Correct.

- Combine the coefficients separately from the variable parts. - An unwritten coefficient is \(1\). - Use a common denominator for fractional coefficients.

1. For part a, \(-1.2+2.5-0.8=0.5\), so the result is \(0.5k\). 2. For part b, \(\frac{2}{3}-\frac{1}{6}-1=\frac{4}{6}-\frac{1}{6}-\frac{6}{6}=-\frac{1}{2}\), so the result is \(-\frac{1}{2}z\). 3. For part c, \(1.5-2+0.5=0\), so the expression equals \(0\). 4. For part d, \(\frac{3}{4}+\frac{1}{8}-1=\frac{6}{8}+\frac{1}{8}-\frac{8}{8}=-\frac{1}{8}\), so the result is \(-\frac{1}{8}ab\).

a) \(0.5k\) b) \(-\frac{1}{2}z\) c) \(0\) d) \(-\frac{1}{8}ab\)

- Group terms with identical powers of \(x\). - Keep each coefficient’s sign. - Combine the cubic and linear terms separately.

1. Combine the \(x^3\)-terms: \(0.7-0.5-0.4=-0.2\). 2. Combine the \(x\)-terms: \(-1.2+2.5=1.3\). 3. The simplified expression is \(-0.2x^3+1.3x\).

\(-0.2x^3+1.3x\)

- Group terms with matching variables and exponents. - Combine constants separately. - Keep the signs attached to their terms.

1. Combine the \(x^2\)-terms: \(4.8x^2-1.3x^2=3.5x^2\). 2. Combine the \(x\)-terms: \(-2.5x-0.5x=-3x\). 3. Combine the constants: \(3.7-1.2=2.5\). 4. The simplified expression is \(3.5x^2-3x+2.5\).

\(3.5x^2-3x+2.5\)

- Simplify Expression A before comparing. - \(a^2\), \(ab\), and \(b\) are different variable parts. - Compare every coefficient and sign after simplifying.

1. Combine the \(a^2\)-terms in Expression A: \(1.5a^2-0.5a^2=a^2\). 2. Combine the \(ab\)-terms: \(-2.4ab+3.1ab=0.7ab\). 3. The \(-0.6b\) term remains unchanged. 4. Expression A simplifies to \(a^2+0.7ab-0.6b\), which is identical to Expression B.

Yes. The expressions are equivalent because Expression A simplifies to \(a^2+0.7ab-0.6b\).

- \(u^2v\) and \(uv^2\) are not like terms. - Use common denominators for the fractional coefficients. - Keep each negative sign with its coefficient.

1. Combine the \(u^2v\)-terms: \(-\frac{4}{9}-\frac{1}{3}=-\frac{4}{9}-\frac{3}{9}=-\frac{7}{9}\). 2. Combine the \(uv^2\)-terms: \(\frac{1}{2}-\frac{3}{10}+\frac{1}{5}=\frac{5}{10}-\frac{3}{10}+\frac{2}{10}=\frac{2}{5}\). 3. The simplified expression is \(-\frac{7}{9}u^2v+\frac{2}{5}uv^2\).

\(-\frac{7}{9}u^2v+\frac{2}{5}uv^2\)

- Group terms with identical variable parts. - Combine the signed decimal coefficients. - Keep the three variable parts separate.

1. Combine the \(m^2\)-terms: \(7.5-5.8=1.7\). 2. Combine the \(mn\)-terms: \(-4.2+2.3=-1.9\). 3. Combine the \(n^2\)-terms: \(1.6-3.1=-1.5\). 4. The simplified expression is \(1.7m^2-1.9mn-1.5n^2\).

\(1.7m^2-1.9mn-1.5n^2\)

- A plus sign before parentheses leaves the signs inside unchanged. - Combine only terms with matching variables and exponents. - Combine constants separately.

1. Combine matching variable terms: \(12a+5a=17a\) and \(7b-9b=-2b\). The result is \(17a-2b\). 2. Combine by degree: \(4x^2+2x^2=6x^2\), \(-3x+5x=2x\), and \(2-6=-4\). The result is \(6x^2+2x-4\). 3. Combine the \(y\)-terms and constants: \(2.4y+0.6y=3y\) and \(-1.5+3.8=2.3\). The result is \(3y+2.3\).

1) \(17a-2b\) 2) \(6x^2+2x-4\) 3) \(3y+2.3\)

- Combine only terms with identical variables and exponents. - Opposite terms cancel. - Group each variable part before adding coefficients.

1. Combine matching variable parts: \(5r^2s-4r^2s=r^2s\) and \(-2rs^2+3rs^2=rs^2\). The result is \(r^2s+rs^2\). 2. The \(a^3\)-terms cancel, \(4a^2+2a^2=6a^2\), and the constants cancel. The result is \(6a^2\). 3. \(2x+x=3x\), \(3y-5y+2y=0\), and \(-4z+2z+2z=0\). The result is \(3x\).

1) \(r^2s+rs^2\) 2) \(6a^2\) 3) \(3x\)

- Group terms with matching variable parts. - Use common denominators for fractional coefficients and constants. - Opposite terms cancel.

1. \((9y-12)+(4y+15)=13y+3\). 2. \((-3a+8b)+(3a-10b)=-2b\). 3. \((2.5x^2-4x)+(-1.5x^2+4x-7)=x^2-7\). 4. \(\left(\frac{1}{2}k+\frac{2}{3}\right)+\left(\frac{1}{4}k-\frac{1}{6}\right)=\frac{3}{4}k+\frac{1}{2}\).

1) \(13y+3\) 2) \(-2b\) 3) \(x^2-7\) 4) \(\frac{3}{4}k+\frac{1}{2}\)

- Group terms with identical variable parts. - Add the signed decimal coefficients. - Keep \(a^2\), \(ab\), and \(b^2\) separate.

1. Combine the \(a^2\)-coefficients: \(7.2-2.8+0.6=5\). 2. Combine the \(ab\)-coefficients: \(-3.5+4.2-1.1=-0.4\). 3. Combine the \(b^2\)-coefficients: \(1.4-0.9-1.2=-0.7\). 4. The sum is \(5a^2-0.4ab-0.7b^2\).

\(5a^2-0.4ab-0.7b^2\)

- Treat the unknown polynomial as a missing addend in a polynomial equation. - Be careful with every sign when isolating the missing polynomial. - Combine terms only when they have the same degree.

1. Let the missing polynomial be \(Q\): \((2x^2+4x-1)+Q=5x^2-3x+8\). 2. Subtract the known polynomial: \(Q=(5x^2-3x+8)-(2x^2+4x-1)\). 3. Distribute the negative sign: \(Q=5x^2-3x+8-2x^2-4x+1\). 4. Combine like terms: \(Q=3x^2-7x+9\).

\(3x^2-7x+9\)

- Treat each blank as an unknown expression. - Pay attention to subtraction of negative terms. - Substitute each result to verify it.

1. Let the missing expression be \(M\). Then \(M=4.5z^2-7z^2=-2.5z^2\). Check: \(4.5z^2-(-2.5z^2)=7z^2\). 2. Let the missing expression be \(N\). Then \(N+3x^3=-x^3\), so \(N=-4x^3\). 3. Combine like terms: \(-1.2a-0.8a=-2a\).

1) \(-2.5z^2\) 2) \(-4x^3\) 3) \(-2a\)

- Combine the numerical coefficients. - Subtracting a negative coefficient changes to addition. - Keep the variable part unchanged.

1. \(5.4-2.1=3.3\), so the result is \(3.3k^2\). 2. \(-3.8-(-5.2)=-3.8+5.2=1.4\), so the result is \(1.4x^4\). 3. \(0.25-0.75=-0.5\), so the result is \(-0.5m^2n\).

1) \(3.3k^2\) 2) \(1.4x^4\) 3) \(-0.5m^2n\)

- Convert fractional and decimal coefficients to a common form. - \(a^2b\) and \(ab^2\) are different variable parts. - A coefficient of \(-1\) is written as a minus sign.

1. Convert \(\frac{3}{4}\) to \(0.75\): \(0.75-0.5-1.25=-1\). The result is \(-y^3\). 2. Combine the \(a^2b\)-terms: \(6.7-4.7=2\). Combine the \(ab^2\)-terms: \(-2.3+0.3=-2\). The result is \(2a^2b-2ab^2\).

1) \(-y^3\) 2) \(2a^2b-2ab^2\)

- Combine only terms with identical variable parts. - Subtracting a negative term changes to addition. - Use common denominators for fractional coefficients.

1. \(12-15+4=1\), so the result is \(xy\). 2. \(-5-(-8)-3=-5+8-3=0\), so the expression equals \(0\). 3. \(4.5+1.5-7=-1\), so the result is \(-z\). 4. \(\frac{1}{2}+\frac{3}{2}=2\), so the result is \(2x\).

1) \(xy\) 2) \(0\) 3) \(-z\) 4) \(2x\)

- A negative sign before parentheses reverses every sign inside. - Combine only matching variable parts. - Keep \(u^2\)-terms separate from \(u\)-terms.

1. For part a, distribute the negative sign: \(9k-4k-3m=5k-3m\). 2. For part b, distribute the negative sign and combine like terms: \(2x-8y-5x+3y=-3x-5y\). 3. For part c, \(15u^2-7u^2+10u-4u=8u^2+6u\).

a) \(5k-3m\) b) \(-3x-5y\) c) \(8u^2+6u\)

- Reverse every sign in the expression being subtracted. - Group terms with identical variable parts. - Check whether any terms cancel.

1. For part a, distribute the negative sign: \(6x-9y+4z-15x-3y+8z\). 2. Combine like terms: \((6-15)x+(-9-3)y+(4+8)z=-9x-12y+12z\). 3. For part b, distribute the negative sign: \(4a^2-ab+2b^2-4a^2-5ab+3b^2\). 4. Combine like terms: the \(a^2\)-terms cancel, \(-ab-5ab=-6ab\), and \(2b^2+3b^2=5b^2\).

a) \(-9x-12y+12z\) b) \(-6ab+5b^2\)

- Reverse every sign in the second polynomial. - Combine terms with identical variable parts. - Use common denominators for fractional coefficients.

1. Distribute the negative sign: \(\frac{3}{10}x^2-\frac{2}{3}xy+\frac{5}{6}y^2-\frac{1}{5}x^2-\frac{1}{4}xy+\frac{1}{2}y^2\). 2. Combine the \(x^2\)-terms: \(\frac{3}{10}-\frac{1}{5}=\frac{1}{10}\). 3. Combine the \(xy\)-terms: \(-\frac{2}{3}-\frac{1}{4}=-\frac{8}{12}-\frac{3}{12}=-\frac{11}{12}\). 4. Combine the \(y^2\)-terms: \(\frac{5}{6}+\frac{1}{2}=\frac{4}{3}\). 5. The result is \(\frac{1}{10}x^2-\frac{11}{12}xy+\frac{4}{3}y^2\).

\(\frac{1}{10}x^2-\frac{11}{12}xy+\frac{4}{3}y^2\)

- Reverse every sign in the second expression. - Group terms with matching variable products. - Combine the signed decimal coefficients.

1. Distribute the negative sign: \(4.7xy-2.3yz+5xz-1.2xy-3.7yz+2xz\). 2. Combine the \(xy\)-terms: \(4.7xy-1.2xy=3.5xy\). 3. Combine the \(yz\)-terms: \(-2.3yz-3.7yz=-6yz\). 4. Combine the \(xz\)-terms: \(5xz+2xz=7xz\). 5. The result is \(3.5xy-6yz+7xz\).

\(3.5xy-6yz+7xz\)

- Convert all coefficients to decimals first. - Reverse every sign in the second polynomial. - Combine the \(x^2\), \(xy\), and \(y^2\) terms separately.

1. Convert the fractional coefficients: \(4\frac{1}{2}=4.5\), \(\frac{4}{5}=0.8\), and \(3\frac{1}{2}=3.5\). 2. Distribute the negative sign: \(4.5x^2-0.3xy+2y^2-1.75x^2-0.8xy+3.5y^2\). 3. Combine like terms: \((4.5-1.75)x^2+(-0.3-0.8)xy+(2+3.5)y^2\). 4. The result is \(2.75x^2-1.1xy+5.5y^2\).

\(2.75x^2-1.1xy+5.5y^2\)

- Convert the fraction and mixed number to decimals. - Reverse every sign in the second polynomial. - Check whether the constant terms cancel.

1. Convert the fractional values: \(\frac{3}{8}=0.375\) and \(1\frac{1}{5}=1.2\). 2. Distribute the negative sign: \(0.6z^2-0.375z+1.2+1.4z^2-0.125z-1.2\). 3. Combine like terms: \((0.6+1.4)z^2+(-0.375-0.125)z+(1.2-1.2)\). 4. The result is \(2z^2-0.5z\).

\(2z^2-0.5z\)

- Convert mixed numbers to improper fractions. - Reverse every sign in the expression being subtracted. - Use common denominators before combining coefficients.

1. Convert \(1\frac{3}{4}\) to \(\frac{7}{4}\), distribute the negative sign, and combine like terms: \(\left(\frac{7}{4}-\frac{1}{2}\right)x+\left(-\frac{2}{5}-\frac{1}{10}\right)y=\frac{5}{4}x-\frac{1}{2}y\). 2. Distribute the negative sign and combine like terms: \(\left(\frac{2}{3}-\frac{1}{6}\right)a^2+\left(-\frac{1}{4}-\frac{3}{4}\right)ab+\left(\frac{1}{2}+1\right)b^2\). 3. The second result is \(\frac{1}{2}a^2-ab+\frac{3}{2}b^2\).

1) \(\frac{5}{4}x-\frac{1}{2}y\) 2) \(\frac{1}{2}a^2-ab+\frac{3}{2}b^2\)

- Begin with the innermost parentheses. - Combine like terms inside the brackets before removing them. - Reverse every sign when subtracting the bracketed expression.

1. Remove the inner parentheses: \(10a^2-[4a^2+6a-a^2+2a]\). 2. Combine like terms inside the brackets: \(10a^2-[3a^2+8a]\). 3. Distribute the negative sign outside the brackets: \(10a^2-3a^2-8a\). 4. Combine the quadratic terms to obtain \(7a^2-8a\).

\(7a^2-8a\)

- Reverse every sign in the polynomial being subtracted. - A plus sign before parentheses leaves the signs unchanged. - Group terms by degree.

1. Remove the parentheses: \(4x^2-3x+5-2x^2-5x+7+x^2-4x-1\). 2. Combine the quadratic terms: \((4-2+1)x^2=3x^2\). 3. Combine the linear terms: \((-3-5-4)x=-12x\). 4. Combine the constants: \(5+7-1=11\). 5. The result is \(3x^2-12x+11\).

\(3x^2-12x+11\)

- Reverse every sign in the polynomial being subtracted. - Group terms whose variable parts, including exponents, are identical. - Combine only the coefficients of like terms.

1. Distribute the negative sign before the second polynomial: \(4.5x^2y-2.3xy^2+1.2x^2y^2-1.8x^2y-3.7xy^2+0.5x^2y^2\). 2. Combine the \(x^2y\) terms: \((4.5-1.8)x^2y=2.7x^2y\). 3. Combine the \(xy^2\) terms: \((-2.3-3.7)xy^2=-6xy^2\). 4. Combine the \(x^2y^2\) terms: \((1.2+0.5)x^2y^2=1.7x^2y^2\). 5. The result is \(2.7x^2y-6xy^2+1.7x^2y^2\).

\(2.7x^2y-6xy^2+1.7x^2y^2\)

- Check every sign carefully when removing the subtraction parentheses. - Group terms with exactly the same variable part. - Combine the coefficients of each group separately.

1. Remove the parentheses, reversing the signs in the polynomial being subtracted: \(7.2u^3v^2-4.1uv^3+2.5w-3.8u^3v^2-1.9uv^3+0.4w+1.5uv^3-3.2w\). 2. Combine the \(u^3v^2\) terms: \((7.2-3.8)u^3v^2=3.4u^3v^2\). 3. Combine the \(uv^3\) terms: \((-4.1-1.9+1.5)uv^3=-4.5uv^3\). 4. Combine the \(w\) terms: \((2.5+0.4-3.2)w=-0.3w\). 5. The result is \(3.4u^3v^2-4.5uv^3-0.3w\).

\(3.4u^3v^2-4.5uv^3-0.3w\)

- Group terms with the same variable part. - Reverse every sign in an expression being subtracted. - Only terms with identical variables and exponents are like terms. - Combine the constant terms separately.

1. Substitute the expressions: \((4x^2-7y+2)+(-2x^2+3y-5)+(x^2-y+1)\). Combine like terms: \((4-2+1)x^2+(-7+3-1)y+(2-5+1)=3x^2-5y-2\). 2. Substitute the expressions: \((4x^2-7y+2)-(-2x^2+3y-5)-(x^2-y+1)\). Remove the subtraction parentheses: \(4x^2-7y+2+2x^2-3y+5-x^2+y-1\). Combine like terms: \((4+2-1)x^2+(-7-3+1)y+(2+5-1)=5x^2-9y+6\).

1) \(3x^2-5y-2\) 2) \(5x^2-9y+6\)

- Group terms with identical variables and exponents. - Reverse every sign in an expression being subtracted. - Simplify the expression inside parentheses before subtracting it.

1. For part a, substitute and combine like terms: \((4.5-2.5+1)x^2+(-5+3-1.5)x+(7-4+2)=3x^2-3.5x+5\). 2. For part b, first find \(B+C=(-2.5+1)x^2+(3-1.5)x+(-4+2)=-1.5x^2+1.5x-2\). 3. Subtract this expression from \(A\): \((4.5x^2-5x+7)-(-1.5x^2+1.5x-2)\). 4. Remove the parentheses and combine like terms: \(4.5x^2+1.5x^2-5x-1.5x+7+2=6x^2-6.5x+9\).

a) \(3x^2-3.5x+5\) b) \(6x^2-6.5x+9\)

- Group terms with exactly the same variables and exponents. - Include the sign attached to each coefficient. - Opposite terms cancel.

1. Combine the \(x^2\) terms: \(9x^2-12x^2=-3x^2\). 2. Combine the \(xy\) terms: \(-5xy+5xy=0\). 3. Combine the \(y^2\) terms: \(3y^2-2y^2=y^2\). 4. The simplified expression is \(-3x^2+y^2\).

\(-3x^2+y^2\)

- Distinguish between \(a^2b\) and \(ab^2\); they are not like terms. - Express \(\frac{1}{2}\) as a decimal to combine it with \(3.5\). - Keep the constant term in the final expression.

1. The \(a^2b\) terms are like terms. Since \(\frac{1}{2}=0.5\), their coefficients have sum \(3.5+0.5=4\). 2. The \(ab^2\) terms are like terms, and their coefficients have sum \(-2+4=2\). 3. The constant \(-7\) has no like term. 4. The simplified expression is \(4a^2b+2ab^2-7\).

\(4a^2b+2ab^2-7\)

- Identify terms with exactly the same variables and exponents. - Group like terms before adding their coefficients. - Include the sign attached to each coefficient. - Combine only the numerical coefficients.

1. Combine the \(x^2\) terms: \((2.4+0.8)x^2=3.2x^2\). 2. Combine the \(xy\) terms: \((-1.5+0.15)xy=-1.35xy\). 3. The term \(-3.2y^2\) has no like term. 4. The simplified expression is \(3.2x^2-1.35xy-3.2y^2\).

\(3.2x^2-1.35xy-3.2y^2\)

- Compare the exponents carefully; \(a^2b\) and \(ab^2\) are not like terms. - A term with no written coefficient has coefficient \(1\). - Group terms with identical variable parts. - Keep constants separate from variable terms.

1. The \(a^2b\) terms are like terms. Their coefficients have sum \(1+1.2=2.2\). 2. The \(ab^2\) terms are like terms. Their coefficients have sum \(-0.4+0.15=-0.25\). 3. The constant \(-0.75\) has no like term. 4. The simplified expression is \(2.2a^2b-0.25ab^2-0.75\).

\(2.2a^2b-0.25ab^2-0.75\)

- Group terms with exactly the same variable and exponent. - Include the sign attached to each coefficient. - Opposite terms cancel. - Arrange like terms next to one another before combining them.

1. Combine the \(x^3\) terms: \(12x^3-5x^3=7x^3\). 2. Combine the \(x^2\) terms: \(-7x^2+7x^2=0\). 3. The terms \(4x\) and \(-9\) have no like terms. 4. The simplified expression is \(7x^3+4x-9\).

\(7x^3+4x-9\)

- Equivalent expressions have the same value for every allowed value of the variable. - Simplify the more complicated expression first. - Group all \(b^2\) terms, all \(b\) terms, and all constants. - Compare the simplified result with \(T_2\).

1. Combine the \(b^2\) terms in \(T_1\): \(6b^2-2b^2=4b^2\). 2. Combine the \(b\) terms: \(-4b+5b=b\). 3. Combine the constants: \(3-8=-5\). 4. Therefore, \(T_1=4b^2+b-5\), which is identical to \(T_2\). 5. The expressions are equivalent because they have the same value for every value of \(b\).

Yes. Simplifying \(T_1\) gives \(4b^2+b-5\), which is identical to \(T_2\).

- Terms are like terms only when all variable bases and exponents match. - A term without a written coefficient has coefficient \(1\). - Track subtraction signs carefully.

1. \(4x^2 + x^2 - 2x^2 = 3x^2\). 2. \(a^2b + a^2b + ab^2 = 2a^2b + ab^2\). 3. \(3z^3 + z^3 - 2z^3 = 2z^3\). 4. \(2p^2 + q^3 + p^2 + q^3 + p^2 = 4p^2 + 2q^3\).

1) \(3x^2\) 2) \(2a^2b + ab^2\) 3) \(2z^3\) 4) \(4p^2 + 2q^3\)

- Group terms with the same variables and exponents. - Adding a negative term is the same as subtracting it. - Combine the coefficients of each group separately. - A term with coefficient \(0\) disappears.

1. Combine the \(x^2\) terms: \((9-5-3)x^2=x^2\). 2. Combine the \(y\) terms: \((-4+4)y=0\). 3. The simplified expression is \(x^2\).

\(x^2\)

- Combine only terms with exactly the same variable part. - Track the signs when adding decimal coefficients. - The terms \(ab\) and \(a^2\) are not like terms. - A group whose coefficients sum to \(0\) disappears.

1. Combine the \(ab\) terms: \((1.8-0.8)ab=ab\). 2. Combine the \(a^2\) terms: \((-0.6-1.4+2)a^2=0\). 3. The simplified expression is \(ab\).

\(ab\)

- Reverse every sign in an expression being subtracted. - Combine only terms with exactly the same variable part. - Distinguish variable terms from terms with powers, such as \(a\) and \(a^2\). - Group like terms before combining their coefficients.

1. For part a, remove the parentheses and combine like terms: \(8x+12y-5x+3y=3x+15y\). 2. For part b, remove the parentheses and combine like terms: \(4a^2-7a-a^2-2a=3a^2-9a\). 3. For part c, remove the parentheses and combine like terms: \(9uv+4v-9uv+6v=10v\). 4. For part d, remove the parentheses and combine like terms: \(-3k+5+7k-2=4k+3\).

a) \(3x+15y\) b) \(3a^2-9a\) c) \(10v\) d) \(4k+3\)

- Distribute the factor outside parentheses to every term inside. - Pay close attention when a product or set of parentheses is subtracted. - Look for terms that cancel.

1. For a), \(5rs - 10r - 3rs - 12s + 10r = 2rs - 12s\). 2. For b), \(2x^2 - 3xy - x^2 + 2xy = x^2 - xy\). 3. For c), \(k^2m - (k^2m - k^2) + m = k^2m - k^2m + k^2 + m = k^2 + m\).

a) \(2rs - 12s\) b) \(x^2 - xy\) c) \(k^2 + m\)

- \(a^2b\) and \(ab^2\) are different variable parts. - Express fractional and decimal coefficients in a common form. - Keep the constant term in the final expression.

1. Combine the \(a^2b\)-terms: \(\frac{2}{5}-0.6=0.4-0.6=-0.2=-\frac{1}{5}\). 2. Combine the \(ab^2\)-terms: \(\frac{3}{2}-\frac{3}{4}=\frac{6}{4}-\frac{3}{4}=\frac{3}{4}\). 3. Keep the constant term \(5\). The result is \(-\frac{1}{5}a^2b+\frac{3}{4}ab^2+5\).

\(-\frac{1}{5}a^2b+\frac{3}{4}ab^2+5\)

- Combine terms with the same powers of \(x\). - Identify the extra term in the simplified expression. - Add its opposite to make it cancel.

1. Combine the quadratic terms: \(-4x^2-2x^2+7x^2=x^2\). 2. Combine the linear terms: \(9x-12x=-3x\). 3. Therefore, \(T=x^2-3x\). 4. To obtain \(x^2\), add the opposite of \(-3x\), which is \(3x\): \(x^2-3x+3x=x^2\).

a) \(x^2-3x\) b) \(3x\)

- Convert every coefficient to the same numerical form. - \(x^2\)-terms and \(x\)-terms are not like terms. - Combine constants separately.

1. Convert the coefficients: \(\frac{3}{4}=0.75\) and \(1\frac{1}{2}=1.5\). 2. Combine the \(x^2\)-terms: \(1.2x^2-0.8x^2=0.4x^2\). 3. Combine the \(x\)-terms: \(-0.75x+1.5x=0.75x\). 4. Combine the constants: \(0.5-1.1=-0.6\). 5. The simplified expression is \(0.4x^2+0.75x-0.6\).

\(0.4x^2+0.75x-0.6\)

- Put the entire polynomial being subtracted in parentheses. - Reverse every sign in the second polynomial. - Compare corresponding coefficients in the two results.

1. \(P-Q=(12x^2+4x-5)-(5x^2-2x+7)\). 2. Distribute the negative sign and combine like terms: \(12x^2+4x-5-5x^2+2x-7=7x^2+6x-12\). 3. \(Q-P=(5x^2-2x+7)-(12x^2+4x-5)\). 4. Simplify: \(5x^2-2x+7-12x^2-4x+5=-7x^2-6x+12\). 5. The second result is the opposite of the first because \(Q-P=-(P-Q)\).

a) \(7x^2+6x-12\) b) \(-7x^2-6x+12\). It is the opposite of the result in part a.

- Distribute the negative sign to every term in the middle polynomial. - Group terms by degree. - Use common denominators for each group of coefficients.

1. Remove the parentheses: \(2a^3-\frac{5}{8}a^2+\frac{1}{3}a-\frac{3}{4}a^3-a^2+\frac{1}{6}a-1+\frac{1}{2}a^3-\frac{1}{4}\). 2. Combine the cubic terms: \(2-\frac{3}{4}+\frac{1}{2}=\frac{7}{4}\). 3. Combine the quadratic terms: \(-\frac{5}{8}-1=-\frac{13}{8}\). 4. Combine the linear terms: \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\). 5. Combine the constants: \(-1-\frac{1}{4}=-\frac{5}{4}\). 6. The simplified polynomial is \(\frac{7}{4}a^3-\frac{13}{8}a^2+\frac{1}{2}a-\frac{5}{4}\).

\(\frac{7}{4}a^3-\frac{13}{8}a^2+\frac{1}{2}a-\frac{5}{4}\)

- Convert all coefficients to a common form. - \(a^2b\) and \(ab^2\) are not like terms. - Remove the parentheses before combining coefficients.

1. Convert the fractional coefficients: \(\frac{1}{2}=0.5\) and \(\frac{1}{4}=0.25\). 2. Remove the parentheses: \(0.5a^2b-0.3ab^2+1.5a^2b-0.7ab^2+0.25a^2b\). 3. Combine the \(a^2b\)-terms: \(0.5+1.5+0.25=2.25\). 4. Combine the \(ab^2\)-terms: \(-0.3-0.7=-1\). 5. The simplified expression is \(2.25a^2b-ab^2\).

\(2.25a^2b-ab^2\)

- Work from the inner parentheses outward. - Simplify the entire bracketed expression before subtracting it. - Reverse every sign when removing the outer brackets.

1. Simplify inside the brackets: \(3y^2-2x^2-5xy+y^2+x^2-5xy\). 2. Combine like terms inside the brackets: \(-x^2-10xy+4y^2\). 3. Subtract the bracketed expression: \(4x^2-[-x^2-10xy+4y^2]=4x^2+x^2+10xy-4y^2\). 4. The result is \(5x^2+10xy-4y^2\).

\(5x^2+10xy-4y^2\)

- Work from the innermost grouping symbols outward. - Pay close attention to each negative sign before a grouped expression. - Treat \(a^2\) terms and \(a\) terms as different kinds of terms. - Combine like terms whenever a grouped expression has been simplified.

1. Simplify the innermost grouping: \(2a-(3a^2-a)=2a-3a^2+a=3a-3a^2\). 2. Substitute into the expression in braces: \(5a^2-[3a-3a^2]+4a\). 3. Remove the brackets and combine like terms: \(5a^2-3a+3a^2+4a=8a^2+a\). 4. Subtract the expression in braces: \(10a^2-(8a^2+a)=2a^2-a\).

\(2a^2-a\)

- Represent the unknown second expression with a variable. - Use the relationship between a difference, its first expression, and its second expression. - Distribute the negative sign carefully before combining like terms.

1. Let the second expression be \(S\). Then \((7a^2-2)-S=3a^2-5a-4\). 2. Solve for \(S\): \(S=(7a^2-2)-(3a^2-5a-4)\). 3. Distribute the negative sign: \(S=7a^2-2-3a^2+5a+4\). 4. Combine like terms: \(S=4a^2+5a+2\). 5. Therefore, \(3a^2-5a-4=(7a^2-2)-(4a^2+5a+2)\).

\((7a^2-2)-(4a^2+5a+2)\)

- Add the two known expressions first. - Subtract their sum from the target polynomial. - Combine only terms with exactly the same variables and exponents.

1. Let the third expression be \(T\). Then \((2a^2b+5)+(-ab^2-3a)+T=a^2b-3ab^2+4a-7\). 2. Combine the known expressions: \(2a^2b-ab^2-3a+5+T=a^2b-3ab^2+4a-7\). 3. Subtract the known sum from the target polynomial: \(T=(a^2b-3ab^2+4a-7)-(2a^2b-ab^2-3a+5)\). 4. Distribute the negative sign: \(T=a^2b-3ab^2+4a-7-2a^2b+ab^2+3a-5\). 5. Combine like terms: \(T=-a^2b-2ab^2+7a-12\).

\(-a^2b-2ab^2+7a-12\)

- Work from the innermost grouping symbols outward. - Pay close attention to the signs when removing parentheses. - Simplify the algebraic expression before substituting the values. - Substitute \(-2\) for \(b\), including its negative sign.

1. Simplify inside the brackets: \(3ab-(a^2+2ab)=3ab-a^2-2ab=ab-a^2\). 2. Simplify the full expression: \(2a^2-(ab-a^2)=2a^2-ab+a^2=3a^2-ab\). 3. Substitute \(a=5\) and \(b=-2\): \(3\cdot5^2-5\cdot(-2)=75+10=85\).

\(85\)

- First simplify using the symbols \(U\) and \(V\). - Track the signs carefully when removing parentheses. - After substituting, group the \(m^2\) terms and the \(n\) terms separately.

1. Simplify \(W\) using \(U\) and \(V\): \(2[U-V+U]-3U=2[2U-V]-3U\). 2. Distribute and combine like terms: \(4U-2V-3U=U-2V\). 3. Substitute the given expressions: \((3m^2-5n)-2(m^2+2n)\). 4. Distribute and combine like terms: \(3m^2-5n-2m^2-4n=m^2-9n\).

\(m^2-9n\)

- Work from the innermost grouping symbols outward. - Reverse every sign when subtracting a grouped expression. - Simplify the algebraic expression before substituting the values. - Remember that the square of a negative number is positive.

1. Remove the inner parentheses: \(2a^2b-[3ab^2-4a^2b+2ab^2+5a^2b]\). 2. Combine like terms inside the brackets: \(2a^2b-[5ab^2+a^2b]\). 3. Remove the brackets and combine like terms: \(2a^2b-5ab^2-a^2b=a^2b-5ab^2\). 4. Substitute \(a=-2\) and \(b=3\): \((-2)^2\cdot3-5\cdot(-2)\cdot3^2\). 5. Evaluate: \(4\cdot3-5\cdot(-2)\cdot9=12+90=102\).

\(102\)

- Work from the innermost grouping symbols outward. - Group like terms, such as all \(x^2\) terms, before combining them. - The product of two negative numbers is positive. - Check every sign after removing a grouped expression.

1. Remove the innermost parentheses: \(4xy-\{2x^2-[3xy-x^2+2xy+4x^2]-5xy\}\). 2. Combine like terms inside the brackets: \(4xy-\{2x^2-[5xy+3x^2]-5xy\}\). 3. Remove the brackets: \(4xy-\{2x^2-5xy-3x^2-5xy\}\). 4. Combine like terms inside the braces: \(4xy-\{-x^2-10xy\}\). 5. Remove the braces and combine like terms: \(4xy+x^2+10xy=x^2+14xy\). 6. Substitute \(x=-1\) and \(y=-4\): \((-1)^2+14\cdot(-1)\cdot(-4)=1+56=57\).

\(57\)

- Work from the innermost grouping symbols outward. - Reverse every sign when subtracting a grouped expression. - Simplify completely before substituting the values. - Evaluate exponents before multiplication.

1. Remove the innermost parentheses: \(x^2y-\{2xy^2-[3x^2y-xy^2-x^2y]\}\). 2. Combine like terms inside the brackets: \(x^2y-\{2xy^2-[2x^2y-xy^2]\}\). 3. Remove the brackets: \(x^2y-\{2xy^2-2x^2y+xy^2\}\). 4. Combine like terms inside the braces: \(x^2y-\{3xy^2-2x^2y\}\). 5. Remove the braces and combine like terms: \(x^2y-3xy^2+2x^2y=3x^2y-3xy^2\). 6. Substitute \(x=-1\) and \(y=\frac{2}{3}\): \(3\cdot(-1)^2\cdot\frac{2}{3}-3\cdot(-1)\cdot\left(\frac{2}{3}\right)^2\). 7. Evaluate: \(2+\frac{4}{3}=\frac{10}{3}\).

\(\frac{10}{3}\), or \(3\frac{1}{3}\)

- Compare each original term with the terms Lucas appears to have used. - What happens when a minus sign stands before parentheses? - Check the linear terms separately from the squared terms.

1. Lucas subtracted \(9x\) instead of subtracting \(-9x\) from the linear term. 2. Removing the parentheses correctly gives \(15x^2-4x-7x^2+9x\). 3. Combining like terms gives \(8x^2+5x\).

Lucas did not change the sign of \(-9x\) when subtracting the second polynomial. The correct result is \(8x^2+5x\).