Skip to main content

Section 10.6 Additional Practice Related to Polynomials

Exercises Exercises

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

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