- The divisor applies to each term, including the constant. - Keep each term's sign while dividing.

1. Divide each term separately: \(12x^2\div4=3x^2\), \(-8x\div4=-2x\), and \(4\div4=1\). 2. The quotient is \(3x^2-2x+1\).

\(3x^2-2x+1\)

- Think about the number part and the variable part separately. - What does \(x^2\) mean as a product of factors?

1. Divide the coefficients: \(12\div4=3\). 2. Since \(x\ne0\), cancel one factor of \(x\) from \(x^2\div x\), leaving \(x\). 3. The quotient is \(3x\).

\(3x\) for \(x\ne0\); cancel \(4\) from the numeric factors and one common factor of \(x\).

- What operation reverses division by \(x\)? - Check the powers of \(x\) in the quotient and the numerator.

1. Multiplying the stated quotient by the divisor gives \(4x\cdot x=4x^2\). 2. Since \(x\ne0\), \(\frac{4x^2}{x}=4x\), so the missing numerator is \(4x^2\).

\(4x^2\). Multiplying \(4x\) by \(x\) gives the numerator, and dividing by \(x\) returns \(4x\).

- Divide every term in the polynomial by the divisor. - Subtract exponents when dividing powers with the same nonzero base. - Apply the sign rules carefully.

1. Divide each term by \(2.5\): \(15b^3 \div 2.5 = 6b^3\), \(-10b^2 \div 2.5 = -4b^2\), and \(5b \div 2.5 = 2b\). The result is \(6b^3 - 4b^2 + 2b\). 2. Divide each term by \(-4a^2\): \(3a^2\), \(-2a\), and \(1\). The result is \(3a^2 - 2a + 1\).

1) \(6b^3 - 4b^2 + 2b\) 2) \(3a^2 - 2a + 1\)

- Dividing by a fraction is equivalent to multiplying by its reciprocal. - Divide every term by the monomial. - Subtract exponents on the common variable base.

1. Divide each coefficient by \(\frac{6}{5}\) and subtract \(2\) from each exponent of \(z\): \(-10z^4 + 15z^2 - 5\). 2. Divide each coefficient by \(-\frac{1}{4}\) and subtract \(1\) from each exponent of \(x\): \(-3x^2 + 2x - 4\).

1) \(-10z^4 + 15z^2 - 5\) 2) \(-3x^2 + 2x - 4\)

- Divide each term separately. - A variable with no written exponent has exponent \(1\). - Check by multiplying your result by the divisor.

1. Divide each term by \(-6a\): \(-12a^3 \div (-6a) = 2a^2\), \(18a^2 \div (-6a) = -3a\), and \(-6a \div (-6a) = 1\). 2. The result is \(2a^2 - 3a + 1\).

\(2a^2 - 3a + 1\)

- Divide every term in the polynomial by the monomial. - Subtract exponents for each common variable base. - Track the sign of each term.

1. Divide the first term: \(12a^3b^2 \div 3ab^2 = 4a^2\). 2. Divide the second term: \(6a^2b^3 \div 3ab^2 = 2ab\). 3. Divide the third term: \(-3ab^4 \div 3ab^2 = -b^2\). 4. The result is \(4a^2 + 2ab - b^2\).

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

- Divide each term by the monomial. - Subtract exponents when dividing powers with the same nonzero base. - Account for the negative divisor in every term.

1. Divide each coefficient by \(-0.5\): \(-2\), \(5\), and \(-1\). 2. Subtract \(2\) from each exponent of \(p\): \(p^3\), \(p^2\), and \(p\). 3. The result is \(-2p^3 + 5p^2 - p\).

\(-2p^3 + 5p^2 - p\)

- Divide each term in the polynomial by the monomial. - Multiply by the reciprocal when dividing fractional coefficients. - Subtract exponents for each common base.

1. \(6x^4y^3 \div \left(\frac{3}{2}x^2y\right) = 4x^2y^2\). 2. \(-\frac{3}{4}x^3y^2 \div \left(\frac{3}{2}x^2y\right) = -\frac{1}{2}xy\). 3. \(\frac{1}{2}x^2y^4 \div \left(\frac{3}{2}x^2y\right) = \frac{1}{3}y^3\). 4. The result is \(4x^2y^2 - \frac{1}{2}xy + \frac{1}{3}y^3\).

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

- Divide every term by the monomial. - Dividing by a fraction is multiplying by its reciprocal. - Cancel common nonzero variable factors.

1. Divide the first term: \(12a^2b \div \left(\frac{6}{5}ab\right) = 10a\). 2. Divide the second term: \(-18ab^2 \div \left(\frac{6}{5}ab\right) = -15b\). 3. Divide the third term: \(6ab \div \left(\frac{6}{5}ab\right) = 5\). 4. The result is \(10a - 15b + 5\).

\(10a - 15b + 5\)

- Rewrite division as multiplication by a reciprocal if helpful. - Divide every term in the numerator by the divisor. - Track signs carefully when dividing by a negative expression. - Apply the quotient rule for exponents.

1. a) Divide each term by \(4y\): \(12x^2y \div 4y = 3x^2\), \(-8xy \div 4y = -2x\), and \(4y \div 4y = 1\). The result is \(3x^2 - 2x + 1\). 2. b) Divide each term by \(-5ab\): \(-15a^3b^2 \div (-5ab) = 3a^2b\), \(10a^2b \div (-5ab) = -2a\), and \(-5ab \div (-5ab) = 1\). The result is \(3a^2b - 2a + 1\).

a) \(3x^2 - 2x + 1\) b) \(3a^2b - 2a + 1\)

- Rewrite the fraction as a sum of two fractions with the same denominator. - Divide numerical coefficients and cancel common variable factors.

1. Divide the first term: \(\frac{6x^2}{3x}=2x\). 2. Divide the second term: \(\frac{9x}{3x}=3\). 3. The quotient is \(2x+3\), valid for \(x\ne0\).

\(2x+3\) for \(x\ne0\)

- What should happen when you multiply a quotient by its divisor? - Check the part containing \(x\) and the constant part separately.

1. Multiplying Sofia's result by \(4\) gives \(32x+12\), not \(8x+12\). 2. Both terms of \(8x+12\) must be divided by \(4\), giving \(\frac{8x}{4}+\frac{12}{4}=2x+3\). 3. Sofia divided the constant term but did not divide the \(8x\)-term.

Her answer fails the check: \(4(8x+3)=32x+12\). The correct quotient is \(2x+3\). She forgot to divide the \(8x\)-term by \(4\).

- How can you undo a division when the divisor and result are known? - Check each term of your polynomial against the stated divisor.

1. Undo the division by multiplying the quotient by the divisor: \(4x(3x+2)\). 2. Expanding gives \(12x^2+8x\). 3. Checking, \(\frac{12x^2}{4x}+\frac{8x}{4x}=3x+2\) for \(x\ne0\).

The original polynomial is \(12x^2+8x\). Dividing it by \(4x\) returns \(3x+2\) for \(x\ne0\).

- Combine only terms with identical variable parts. - Divide each remaining term by the monomial. - Determine the sign, coefficient, and variable power separately.

1. Combine like terms: \(0.5a^2 - 1.1a^2 = -0.6a^2\). 2. The dividend becomes \(-0.2a^3 - 0.6a^2 + 0.04a\). 3. Divide each term by \(-0.2a\): \(a^2 + 3a - 0.2\).

\(a^2 + 3a - 0.2\)

- Use multiplication as the inverse of division. - Multiply every term in the quotient by the divisor. - Use the given middle term to check the pattern.

1. Multiply each term on the right by \(4xy\). 2. The first missing term is \(2x^2(4xy)=8x^3y\). 3. The middle term checks: \(-3xy(4xy)=-12x^2y^2\). 4. The final missing term is \(5y^2(4xy)=20xy^3\). 5. Thus, \(\frac{8x^3y-12x^2y^2+20xy^3}{4xy}=2x^2-3xy+5y^2\).

The missing terms are \(8x^3y\) and \(20xy^3\).