Chapter A.3, Problem 139AYU

To determine

To prove: Whether the degree of the product of two nonzero polynomials equals the sum of their degrees.

Answer to Problem 139AYU

The degree of the product of two nonzero polynomials equals the sum of their degrees.

Explanation of Solution

Given information:.

Calculation:

Calculate the degree of the product of two nonzero polynomials equals the sum of their degrees.

  $\mathrm{deg}\left(uv\right)=\mathrm{deg}\left(u\right)+\mathrm{deg}\left(v\right)$ …. (1)

Where $u$ and $v$ are polynomials.

Consider the two polynomials

  $u=2x^2+5$

  $v=3x^2-4$

First, we will find the multiplication of above two polynomials

  $uv=\left(2x^2+5\right)\left(3x^2-4\right)$

Therefore the multiplication $uv=\left(2x^2+5\right)\left(3x^2-4\right)$ is simplified as

  $\begin{array}{l}uv=2x^2\left(3x^2-4\right)+5\left(3x^2-4\right)\\ uv=6x^4-8x^2+15x^2-20\\ uv=6x^4+7x^2-20\end{array}$

Now we will calculate the degree of $u$ , $v$ and $uv$ and than substitute in equation-(1)

  $\begin{array}{l}\mathrm{deg}\left(uv\right)=4,\\ \mathrm{deg}\left(u\right)=2,\\ \mathrm{deg}\left(v\right)=2\end{array}$

Thus we have

  $\mathrm{deg}\left(uv\right)=\mathrm{deg}\left(u\right)+\left(v\right)=4$

Therefore, it is proved that “the degree of the product of two nonzero polynomials equals the sum of their degrees.”