Q. 378 - Intermediate Algebra

Question

In the following exercise, factor.

$48y^2+ 12y - 36$

Step-by-Step Solution

Answer

The factorisation of the given polynomial is: 

$48y^2+12y-36=12(y+1)(4y-3)$

Step 1. Given information

The given expression is 

$48y^2+12y-36$

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  $48y^2 + 12y - 36$, we have:

Firstly taking out $12$ as a common, we get:

$12(4y^2+y-3)$

Then.

$$\begin{gathered} ac=4\times (-3) \\ =-12 \end{gathered}$$

And we have to think of two numbers whose sum is equal to $1$ and the product is equal to $-12$.

Then,

$$\begin{gathered} 4-3=1 \\ 4\times (-3)=-12 \end{gathered}$$

The required numbers are $4\&-3$

Step 3. Perform factorisation

Now, 

$$\begin{gathered} 12(4y^2+y-3) \\ 12(4y^2+4y-3y-3) \\ 12\left[4y\right(y+1)-3(y+1\left)\right] [taking out 4y and -3 as common] \\ 12(y+1)(4y-3) [taking out (y+1) as common] \end{gathered}$$

The factorisation of the given polynomial is: 

$48y^2+ 12y-36=12(y+1)(4y-3)$

Step 4. Check the answer

Multiplying the factors, we get:

$$\begin{gathered} 48y^2+ 12y-36=12(y+1)(4y-3) \\ 48y^2+ 12y-36=12(4y^2+y-3) \\ 48y^2+ 12y-36=48y^2+ 12y-36 \end{gathered}$$

This is true