Chapter 4.2, Problem 63E

(a)

To determine

To find; the degree of the polynomial $g\left(x\right)+h\left(x\right)$

Answer to Problem 63E

The degree of polynomial $g\left(x\right)+h\left(x\right)$ is 5.

Explanation of Solution

Given information:

Let $g\left(x\right)=12x^4+8x+9$ and $h\left(x\right)=3x^5+2x^3-7x+4$ .

Calculation:

Evaluate the function $g\left(x\right)+h\left(x\right)$ as,

  $\begin{array}{c}g\left(x\right)+h\left(x\right)=12x^4+8x+9+3x^5+2x^3-7x+4\\ =3x^5+12x^4+2x^3+\left(8x-7x\right)+\left(9+4\right)\\ =3x^5+12x^4+2x^3+x+13\end{array}$

Because the highest exponent is 5, so the degree of polynomial $g\left(x\right)+h\left(x\right)$ is 5.

(b)

To determine

To find; the degree of the polynomial $g\left(x\right)-h\left(x\right)$

Answer to Problem 63E

The degree of polynomial $g\left(x\right)-h\left(x\right)$ is 5.

Explanation of Solution

Given information:

Let $g\left(x\right)=12x^4+8x+9$ and $h\left(x\right)=3x^5+2x^3-7x+4$ .

Calculation:

Evaluate the function $g\left(x\right)-h\left(x\right)$ as,

  $\begin{array}{c}g\left(x\right)-h\left(x\right)=12x^4+8x+9-3x^5-2x^3+7x-4\\ =-3x^5+12x^4-2x^3+\left(8x+7x\right)+\left(9-4\right)\\ =-3x^5+12x^4-2x^3+15x+5\end{array}$

Because the highest exponent is 5, so the degree of polynomial $g\left(x\right)-h\left(x\right)$ is 5.

(c)

To determine

To find; the degree of the polynomial $g\left(x\right)·h\left(x\right)$

Answer to Problem 63E

The degree of polynomial $g\left(x\right)·h\left(x\right)$ is 9.

Explanation of Solution

Given information:

Let $g\left(x\right)=12x^4+8x+9$ and $h\left(x\right)=3x^5+2x^3-7x+4$ .

Calculation:

Evaluate the function $g\left(x\right)·h\left(x\right)$ as,

  $\begin{array}{c}g\left(x\right)·h\left(x\right)=\left(12x^4+8x+9\right)\cdot \left(3x^5+2x^3-7x+4\right)\\ =12x^4\left(3x^5+2x^3-7x+4\right)+8x\left(3x^5+2x^3-7x+4\right)\\ +9\left(3x^5+2x^3-7x+4\right)\\ =36x^9+24x^7+24x^6-57x^5+64x^4+18x^3-56x^2-31x+36\end{array}$

Because the highest exponent is 5, so the degree of polynomial $g\left(x\right)·h\left(x\right)$ is 9.

(d)

To determine

To find: the degree of $g\left(x\right)+h\left(x\right),g\left(x\right)-h\left(x\right),\text{and}g\left(x\right)·h\left(x\right)$

Answer to Problem 63E

The degree of $g\left(x\right)+h\left(x\right)$ is $m$ , the degree of $g\left(x\right)-h\left(x\right)$ is $m$ and degree of $g\left(x\right)·h\left(x\right)$ is $m+n$ .

Explanation of Solution

Given information:

Let $g\left(x\right)=12x^4+8x+9$ and $h\left(x\right)=3x^5+2x^3-7x+4$ .

  $g\left(x\right)$ and $h\left(x\right)$ are polynomials such that $g\left(x\right)$ has degree $m$ and $h\left(x\right)$ has degree $n$ , and $m>n$ .

Calculation:

The degree of $g\left(x\right)+h\left(x\right)$ is $m$ , the degree of $g\left(x\right)-h\left(x\right)$ is $m$ and degree of $g\left(x\right)·h\left(x\right)$ is $m+n$ .