Chapter 4.4, Problem 9E

To determine

To factor: the polynomial completely.

Answer to Problem 9E

The factorized form is $2q^4+9q^3-18q^2=q^2\left(2q-3\right)\left(q+6\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $2q^4+9q^3-18q^2$

Calculation:

Consider the given polynomial.

  $2q^4+9q^3-18q^2$

Take out $q^2$ common for each term.

  $2q^4+9q^3-18q^2=q^2\left(2q^2+9q-18\right)$

Now, break the coefficient of middle term $9$ as sum or difference of two numbers such that product of the numbers is equal to the product of first term and constant term that is product of $2$ and $-18$ .

Such numbers will be $12$ and $-3$ .

Now, factorize $2q^2+9q-18$ as shown:

  $\begin{array}{c}2q^2+9q-18=2q^2+12q-3q-18\\ =2q\left(q+6\right)-3\left(q+6\right)\\ =\left(2q-3\right)\left(q+6\right)\end{array}$

Therefore, $2q^4+9q^3-18q^2=q^2\left(2q-3\right)\left(q+6\right)$