Q. 374 - Intermediate Algebra

Question

In the following exercise, factor.

$15p^2 + 2p - 8$

Step-by-Step Solution

Answer

Factorisation of the given polynomial is:

 $15p^2 + 2p - 8=(5p+4)(3p-2)$

Step 1. Given information

The given expression is

$15p^2 + 2p - 8$

Step 2. Use ac method for factorisation

To factorise the polynomial $a{x}^2+bx+c$ by $ac$ method, we need to think of two numbers whose product is equal to $ac$ and the sum is equal to  $b$.

For polynomial  $15p^2+ 2p - 8$ we have:

$$\begin{gathered} ac=15\times (-8) \\ =-120 \end{gathered}$$

And we have to think of two numbers whose sum is equal to $2$ and product is equal to $-120$.

Then,

$$\begin{gathered} 12-10=2 \\ 12\times (-10)=-120 \end{gathered}$$

So the required numbers are $12 \&-10$

Step 3. Perform factorisation

Now, 

$$\begin{gathered} 15p^2+ 2p - 8 \\ 15p^2+12p-10p-8 \\ 3p(5p+4)-2(5p+4) [taking out 3p and -2 as common] \\ (5p+4)(3p-2) [taking out 5p+4 as common] \end{gathered}$$

The factorisation of the given polynomial is:

$(5p+4)(3p-2)$

Step 4. Check the answer

Multiplying the factors, we get:

$$\begin{gathered} 15p^2+ 2p-8=\left(5p+4\right)(3p-2) \\ 15p^2+ 2p-8=5p^2-10p+12p-8 \\ 15p^2+ 2p-8=15p^2+ 2p-8 \end{gathered}$$

This is true.