Chapter A.3, Problem 89AYU

To determine

To find: The factor of the polynomial $y^4+11y^3+30y^2$ and if the polynomial cannot factored at all, say it is prime.

Answer to Problem 89AYU

The factor of the polynomial $y^4+11y^3+30y^2$ is $y^4+11y^3+30y^2=y^2\left(y+6\right)\left(y+5\right)$ .

Explanation of Solution

Given information:

The given polynomial is $y^4+11y^3+30y^2$ .

Calculation:

Write the given polynomial expression as a product consist of only prime factors.

First factor a $y^2$ out of each term of the polynomial $y^4+11y^3+30y^2$ .

  $y^4+11y^3+30y^2=y^2\left(y^2+11y+30\right)$

Calculate the factor for the expression $y^2+11y+30$ .

Write the polynomial in the form $x^2+bx+c$ .

  $\begin{array}{c}{y}^2+11y+30=y^2+6y+5y+30\\ =y\left(y+6\right)+5\left(y+6\right)\\ =\left(y+6\right)\left(y+5\right)\end{array}$

The above equation cannot be factored any further. The polynomial $y^4+11y^3+30y^2$ factored as $y^4+11y^3+30y^2=y^2\left(y+6\right)\left(y+5\right)$ .

Therefore, the factor of the polynomial $y^4+11y^3+30y^2$ is $y^4+11y^3+30y^2=y^2\left(y+6\right)\left(y+5\right)$ .