Chapter 1.3, Problem 133E

  $\left(a\right)$

To determine

To explain: The degree of polynomial when two polynomials are multiplied.

Answer to Problem 133E

When are polynomials are multiplied degrees are added up.

Explanation of Solution

Given information:

The two polynomials.

Formula used:

The law of exponent $a^n\cdot a^m=a^{n+m}$ .

When polynomials are multiplied the degree of the product polynomial is the sum of the degree.

In the resultant polynomial the leading power of both the terms will emerge.

Consider two polynomials say $\left(x^2+x+3\right)$ and $\left(x-1\right)$ .

Now, product of the above two polynomials is,

  $\begin{array}{c}\left(x^2+x+3\right)\left(x-1\right)=x^3-x^2+x^2-x+3x-3\\ =x^3+2x-3\end{array}$

Thus, when are polynomials are multiplied degrees are added up.

  $\left(b\right)$

To determine

To explain: The degree of polynomial when two polynomials are added.

Answer to Problem 133E

When polynomials are added degree is at most the largest of the degrees.

Explanation of Solution

Given information:

The two polynomials.

When polynomials are added the degree is at most the largest of the degrees.

In the resultant polynomial it is not necessary that leading power of both the terms will emerge.

Consider two polynomials say $\left(x^2+x+3\right)$ and $\left(x-1\right)$ .

Now, add the above two polynomials,

  $\begin{array}{c}\left(x^2+x+3\right)+\left(x-1\right)=x^2+x+3+x-1\\ =x^2+2x+2\end{array}$

Thus, polynomials are added degree is at most the largest of the degrees.