- Rewrite a variable in a denominator using exponent notation. - Polynomial exponents on the variable must be nonnegative integers. - The degree comes from the term with the greatest allowed exponent.

1. \(P(x)\) is a polynomial because its exponents on \(x\) are nonnegative integers. 2. Its highest exponent is \(3\), so its degree is \(3\), and the coefficient of the leading term \(4x^3\) is \(4\). 3. \(Q(x)\) is not a polynomial in \(x\) because \(\frac{5}{x}=5x^{-1}\) has a negative exponent.

\(P(x)\) is the polynomial. Degree: \(3\) Leading coefficient: \(4\)

- Expand both products carefully before combining terms. - Keep track of the subtraction in front of \(x(x-10)\). - After simplifying, compare the result with the general linear form \(mx+b\).

1. Expand the square and product: \((x-4)^2=x^2-8x+16\) and \(x(x-10)=x^2-10x\). 2. Substitute and combine like terms: \(f(x)=x^2-8x+16-x^2+10x-15=2x+1\). 3. The quadratic terms cancel, leaving the form \(mx+b\). Therefore, \(f\) is linear.

\(f(x)=2x+1\). It is linear because the simplified expression has the form \(mx+b\).

- Simplify each product before subtracting. - Recognize the conjugate product \((a - 4)(a + 4)\). - Distribute the negative sign across the second result.

1. Distribute: \(3a(2a - 5) = 6a^2 - 15a\). 2. Use the difference of squares: \((a - 4)(a + 4) = a^2 - 16\). 3. Subtract and combine like terms: \(6a^2 - 15a - (a^2 - 16) = 5a^2 - 15a + 16\).

\(5a^2 - 15a + 16\)

- Which exponent property applies when a power is raised to another power? - What happens to the exponents when powers with the same base are divided? - Pay close attention to the signs when distributing \(-3x\).

1. Simplify the power in the numerator: \((2x^3)^2 = 2^2(x^3)^2 = 4x^6\). 2. Divide powers with the same base: \(\frac{4x^6}{x^4} = 4x^{6-4} = 4x^2\). 3. Distribute: \(-3x(x - 2) = -3x^2 + 6x\). 4. Combine like terms: \(4x^2 - 3x^2 + 6x = x^2 + 6x\).

\(x^2 + 6x\)

- Check whether every exponent of \(x\) is a nonnegative integer. - Rewrite divisions and radicals involving \(x\) as powers when helpful. - Polynomial coefficients may be fractions or irrational real numbers. - Expand products to reveal standard polynomial form.

1. A polynomial function has real coefficients and only nonnegative integer exponents of \(x\). 2. For a), \(f_1(x)=\frac{1}{5}x^5-2\), so it is a polynomial. 3. For b), the exponent \(\frac{1}{2}\) is not a nonnegative integer, so it is not a polynomial. 4. For c), \((2x+3)^2=4x^2+12x+9\), so it is a polynomial. 5. For d), \(\frac{3}{x}=3x^{-1}\) has a negative exponent, so it is not a polynomial. 6. For e), \(\sqrt{7}\) is a real coefficient and the exponent \(3\) is a nonnegative integer, so it is a polynomial.

a) Yes: \(f_1(x)=\frac{1}{5}x^5-2\) b) No c) Yes: \(f_3(x)=4x^2+12x+9\) d) No e) Yes: \(f_5(x)=\sqrt{7}x^3\)

- Divide each product term by \(4a^2\) to recover the corresponding term in the polynomial. - Division reverses multiplication. - For part b), distribute \(-2ab\) to every term in the polynomial. - Check signs and exponents carefully.

1. Divide each term by \(4a^2\): \(12a^3 \div 4a^2 = 3a\), \(-8a^2b \div 4a^2 = -2b\), and \(20a^2 \div 4a^2 = 5\). The original polynomial is \(3a - 2b + 5\). 2. Multiply by \(-2ab\): \(-2ab(3a - 2b + 5) = -6a^2b + 4ab^2 - 10ab\).

a) \(3a - 2b + 5\) b) \(-6a^2b + 4ab^2 - 10ab\)

- Use the order of operations. - Divide each term inside the parentheses by \(3\). - A minus sign before parentheses changes both signs when the parentheses are removed. - Combine only terms with the same variable part.

1. Distribute: \(3x(x - 4) = 3x^2 - 12x\). 2. Divide each term: \((6x^2 + 9x) \div 3 = 2x^2 + 3x\). 3. Subtract and combine like terms: \(3x^2 - 12x - (2x^2 + 3x) = x^2 - 15x\).

\(x^2 - 15x\)

- Check the exponents of \(x\). - Expand products before identifying degree and leading coefficient. - Distinguish a power function from an exponential function. - Real coefficients may include fractions and irrational numbers.

1. A polynomial has real coefficients and only nonnegative integer powers of \(x\). 2. For a), the highest power is \(3\), so it is a degree-\(3\) polynomial with leading coefficient \(0.2\). 3. For b), the variable appears in a denominator, so the function is not polynomial. 4. For c), expand: \((x^2-1)(2-x)=-x^3+2x^2+x-2\). It is degree \(3\) with leading coefficient \(-1\). 5. For d), \(3^x\) is exponential, so the function is not polynomial. 6. For e), \(g_5(x)=\frac{1}{8}x^4-\pi x\). It is degree \(4\) with leading coefficient \(\frac{1}{8}\).

a) Polynomial; degree \(3\); leading coefficient \(0.2\) b) Not polynomial c) Polynomial; degree \(3\); leading coefficient \(-1\) d) Not polynomial e) Polynomial; degree \(4\); leading coefficient \(\frac{1}{8}\)

- Recall what restrictions a polynomial places on coefficients and exponents. - For addition and subtraction, ask whether combining like terms can create a forbidden exponent. - For multiplication, use the exponent rule for multiplying powers with the same base. - Consider why starting with finitely many terms matters.

1. A polynomial has finitely many terms with real coefficients and nonnegative integer exponents. 2. When polynomials are added or subtracted, terms are added or subtracted coefficient by coefficient. Real coefficients remain real, and no new exponent type is created. Combining like terms still leaves finitely many polynomial terms. 3. When polynomial terms are multiplied, their real coefficients are multiplied and their nonnegative integer exponents are added. The product coefficient is real, and the sum of nonnegative integers is still a nonnegative integer. 4. Two finite polynomials produce only finitely many partial products. Combining like terms therefore leaves a polynomial. 5. Thus, polynomials are closed under addition, subtraction, and multiplication.

Polynomials are closed under addition, subtraction, and multiplication. Addition and subtraction only combine coefficients of allowed terms, while multiplication multiplies real coefficients and adds nonnegative integer exponents. Because the starting polynomials have finitely many terms, each result also has finitely many valid polynomial terms.

- Focus first on the leading term of each polynomial rather than expanding every term. - Ask whether any other product term could have the same exponent as the product of the two leading terms. - For the sum, combine the highest-degree terms before deciding the degree.

1. The leading terms of \(P\) and \(Q\) are \(3x^4\) and \(-2x^3\). Their product is \(-6x^7\), which cannot be canceled by any other product term. Therefore, \(P(x)Q(x)\) has degree \(7\). 2. \(R(x)+S(x)=(5x^4-5x^4)+3x^3+2x^2+3=3x^3+2x^2+3\), so the sum has degree \(3\). 3. For a product of nonzero polynomials, the leading terms multiply to a nonzero leading term, so the degrees add. In a sum, leading terms of the same degree can cancel, which can lower the degree.

a) Degree \(7\) b) Degree \(3\) c) Degrees add in a product of nonzero polynomials, but leading-term cancellation can lower the degree of a sum.