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Section9.6Additional Practice Related to Polynomials

Subsection9.6.1Exercises

Questions vary.

1

Identify the degree, the leading term, the leading coefficient, the linear term(s), and the constant term for the following polynomial.

\begin{equation*} -2+8x+4x^3-x^4 \end{equation*}
Solution

The degree is \(4\text{,}\) the leading term is \(-x^4\text{,}\) the leading coefficient is \(-1\text{,}\) the linear term is \(8x\text{,}\) and the constant term is \(-2\text{.}\)

2

Identify the degree, the leading term, the leading coefficient, the linear term(s), and the constant term for the following polynomial.

\begin{equation*} 4x+12xy-8y \end{equation*}
Solution

The degree is \(2\text{,}\) the leading term is \(12xy\text{,}\) the leading coefficient is \(12\text{,}\) the linear terms are \(4x\) and \(-8y\text{,}\) and there are no constant terms.

3

Identify each of the following as a trinomial, binomial, monomial, or not any of those type of polynomial.

\begin{equation*} 7xy \end{equation*}
\begin{equation*} 3x^3-2x^2+x-1 \end{equation*}
\begin{equation*} x^2y^2+6 \end{equation*}
\begin{equation*} -x^3+7x^2+8x \end{equation*}
Solution
\begin{equation*} 7xy\text{ is a monomial} \end{equation*}
\begin{equation*} 3x^3-2x^2+x-1\text{ is not a trinomial, binomial, or monomial} \end{equation*}
\begin{equation*} x^2y^2+6\text{ is a binomial} \end{equation*}
\begin{equation*} -x^3+7x^2+8x\text{ is a trinomial} \end{equation*}

Perform the indicated operation and simplify the result.

4

\((-3x^2+5x+4)-(-8x^2-4x+7)\)

Solution

\((-3x^2+5x+4)-(-8x^2-4x+7)=5x^2+9x-3\)

5

\(2(x^4-3x^3+2x-4)+(6x^4-5x^2-4x+8)\)

Solution

\(2(x^4-3x^3+2x-4)+(6x^4-5x^2-4x+8)=8x^4-6x^3-5x^2\)

6

\(-(-3x^2+6x-3)-2(x^2-7x+4)\)

Solution

\(-(-3x^2+6x-3)-2(x^2-7x+4)=x^2+8x-5\)

7

\((4x+3)(2x-9)\)

Solution

\((4x+3)(2x-9)=8x^2-30x-27\)

8

\((2x-9)^2\)

Solution

\((2x-9)^2=4x^2-36x+81\)

9

\(x(x+8)(x-8)\)

Solution

\(x(x+8)(x-8)=x^3-64x\)

10

\((-x^2y^4+2xy)(3x^2y^4-5xy)\)

Solution

\((-x^2y^4+2xy)(3x^2y^4-5xy)=-3x^4y^8+11x^3y^5-10x^2y^2\)

11

\((x+4)(x^2-7x+3)\)

Solution

\((x+4)(x^2-7x+3)=x^3-3x^2-25x+12\)

12

\((x^3+3x^2-x-4)(x-3)\)

Solution

\((x^3+3x^2-x-4)(x-3)=x^4-10x^2-x+12\)

13

\((y+5)^3\)

Solution

\((y+5)^3=y^3+15y^2+75y+125\)

14

\(\frac{-3x^2+15x-3}{-3}\)

Solution

\(\frac{-3x^2+15x-3}{-3}=x^2-5x+1\)

15

\(\frac{42x^5y^8-54x^4y^6+18x^3y^4}{6xy^4}\)

Solution

\(\frac{42x^5y^8-54x^4y^6+18x^3y^4}{6xy^4}=7x^4y^4-9x^3y^2+3x^2\)

16

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

Solution

\(\frac{(xy+3x)(xy-3x)}{-x^2}=-y^2+9\)