Q. 376 - Intermediate Algebra

Question

n the following exercise, factor.

$8a^2+32a+24$

Step-by-Step Solution

Answer

Factorisation of the given polynomial is: 

$8a^2+ 32a+24=8(a+3)(a+1)$

Step 1. Given information

The given expression is 

$8a^2+32a+24$

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 $8a^2+32a +24$we have:  

$$\begin{gathered} ac=24\times 8 \\ =192 \end{gathered}$$

And we have to think of two numbers whose sum is equal to $32$ and product is equal to $192$.

Then,

$$\begin{gathered} 24\times 8=192 \\ 24+8=32 \end{gathered}$$

So, the two required numbers are $24\&8$

Step 3. Perform factorisation

Now,  

$$\begin{gathered} 8a^2+32a+ 24 \\ 8a^2+24a+8a+24 \\ 8a(a+3)+8a(a+3) [taking 8a out as a common] \\ (a+3)(8a+8) [taking (a+3) out as a common] \\ 8(a+3)(a+1) [taking 8 out a common] \end{gathered}$$

The factorisation of the given polynomial is:

$8a^2+ 32a+24=8(a+3)(a+1)$

Step 4. Check the answer

Multiplying the factors, we get:

$$\begin{gathered} 8a^2+ 32a+24=8(a+3)(a+1) \\ 8a^2+ 32a+24=8(a^2+4a+3) \\ 8a^2+ 32a+24=8a^2+ 32a+24 \end{gathered}$$

This is true