Q. 417 - Intermediate Algebra

Question

Recognize and Use the Appropriate Method to Factor a Polynomial Completely.

$5q^2-15q-90$

Step-by-Step Solution

Answer

The factors of the given expression are $5(q-6)(q+3)$. 

Step 1. Given and explanation.

We have $5q^2-15q-90$.

We will first take out the common factors from the equation. Then we will factorize it by ac method where two numbers are need whose product is equal to the product of first and last term and whose sum is the middle term. Then by writing the equation by splitting the middle term will give us the factors.

Step 2. Taking the common factor out an finding numbers.

We have $5q^2-15q-90$.

Taking $5$ common will give,

$5(q^2-3q-18)$.

Now we need to find two numbers whose product is $-18$ and sum is $-3$. Negative sum and product show that the number with larger value is negative. The factors of $-18$ are-

Numbers	Sums
$1 and -18$	$-17$
$2 and -9$	$-7$
$3 and -6$	$-3$

This shows that two numbers are $3,-6$.

Step 3. Splitting middle term.

Splitting the middle term gives us,

 $$\begin{gathered} =5(q^2-3q-18) \\ =5(q^2+3q-6q-18) \\ =5\left[q(q+3)-6(q+3)\right] \\ =5(q+3)(q-6) \end{gathered}$$

This shows that factors are $5(q-6)(q-3)$.

Step 4. Check the solution.

We will check the solution by simply multiplying the factors. If we get the given equation as the product, our answer is right.

So,

$$\begin{gathered} =5(q-6)(q+3) \\ =(5q-30)(q+3) \\ =5q^2+15q-30q-90 \\ =5q^2-15q-90 \end{gathered}$$

Thus our calculations are right.