Chapter 14.7, Problem 18HP

To determine

Explain the polynomial that can befactored. Also, explain the anotherpolynomial that cannot be factored.

Answer to Problem 18HP

The factors of the polynomial $x^2+24x+95$ are $x+19\text{and}x+5$ and the polynomial expression $x^2+x+2$ cannot be factored.

Explanation of Solution

Calculation:

Take a polynomial expression $x^2+24x+95$ .

Split $24x$ into two parts such that their product is equal to the product of $x^2$ and $95$ .

  $\begin{array}{c}{x}^2+24x+95=x^2+5x+19x+95\\ =x\left(x+5\right)+19\left(x+5\right)\\ =\left(x+5\right)\left(x+19\right)\end{array}$

Thus, the factors of the polynomial $x^2+24x+95$ are $x+19\text{and}x+5$ .

Now, take another polynomial expression is $x^2+x+2$ .

In this expression, it is not possible to split $x$ into two parts whose product will be equal to the product of $x^2$ and $2$ .

Hence the polynomial expression $x^2+x+2$ cannot be factored.