Question

Factoring by grouping step by step guide, with explanation. 9x³ + 36x² -4x -16Professor makes us start with Grouping--(1)9x^3+36x^2 : -4x-16than figure out if I am using squares or cubesthen use the a^2-b^2 formula or cubed or sumI have to use these to solve and I am so lost where to began and how to do this

Answer 1

Step 1: Write out the expression

$9x^3+36x^2-4x-16$

Step 2: Group the expressions and factorize

$\begin{gathered} 9x^3+36x^2-4x-16=(9x^3+36x^2)+(-4x-16) \\ Factorize\text{ the expression on the right, to get} \\ 9x^3+36x^2-4x-16=9x^2\mleft(x+4\mright)-4\mleft(x+4\mright) \\ \text{Any expression of the form am - bm can be factorized to becom (a - b)m} \\ \text{ Similarly }9x^2(x+4)-4(x+4)\text{ becomes (}9x^2-4)(x+4) \\ a=9x^2,b=-4,m=x+4 \end{gathered}$

Step 3: Factorize 9x² -4 using the "AC" method

$9x^2-4=9x^2+0x-4$

To use the "AC" method we find the product of the constant -4 and the coefficient of x²,9. The product is -36.

Next, we find two real numbers such that their product is -36 and their sum is the coefficient of x, which in this case is 0.

Consider the real numbers +6 and -6.

Their product is given by

$-6\times(+6)=-36$

Their sum is given by

$-6+6=0$

Hence, we have found our two numbers.

Therefore,

$\begin{gathered} 9x^2+0-4=9x^2+(-6x+6x)-4 \\ =9x^2-6x+6x-4 \\ \text{Grouping the terms in the left, we get} \\ 9x^2+0-4=(9x^2-6x)+(6x-4) \\ =3x(3x-2)+2(3x-2) \\ =(3x+2)(3x-2) \end{gathered}$

Therefore,

$9x^3+36x^2-4x-16=(x+4)(3x+2)(3x-2)$

Hence the polynomial 9x³ + 36x² -4x -16 is factored completely to

(x + 4)(3x + 2)(3x - 2)