Question

Which shows one way to determine the factors of 4x3 + x2 – 8x – 2 by grouping?

Answer 1

Remember that to factoring by grouping, we should split the expression into tow pairs that have something in common. The idea is that after factoring each pair, we get a binomial common factor. Is worth pointing out that you can pair factors in more than one way.

Let's check how to determine the factors of our expression by grouping:

Step 1. Look for common multiples between the terms.

Notice that 4 and 8 are multiples of 2, 8 is a multiple of 4, and all of them are multiples of 1, so we are going to try more than one pairing combination.

Step 2. Look for common factors between the variables.

Usually, we want to pair variables with higher exponents so when we factor each pair, we end up with another common factor in binomial form.

Step 3. Factor each pair; if you end with another binomial common factor, you nailed it. If not, try again.

Les't apply our steps to our expression $4x^3+x^2-8x-2$

Applying step 1, we are pairing $4x^3$ and $-8x$. Remember to pair the remaining factors as well.

$(4x^3-8x)+(x^2-2)$

The common factor in the first pair is $4x$, so lets factor that out

$4x(x^2-2)+(x^2-2)$

Look! we have the binomial common factor $(x^2-2)$. Unfortunately, $4x(x^2-2)+(x^2-2)$ is not in our given choices, so let's try again:

This time we are going to group the higher variables (step 2) $4x^3$ and $x^2$

$4x^3+x^2-8x-2$

$(4x^3+x^2)+(-8x-2)$

The common factor in the first pair is $x^2$, and the common factor in the second one is -2, so let's factor out both of them:

$(4x^3+x^2)+(-8x-2)$

$x^2(4x+1)-2(4x+1)$

This time, we got option A.

We can conclude that the correct answer is: A) x2(4x + 1) – 2(4x + 1)