Step 4: Unlock the Factor by Grouping - Toolbox

Fill in the blanks to complete the factoring process:

Given the polynomial: [tex]$A^4 + 2A^3B - 3A - 6B$[/tex]

Step 1: [tex]$A^{\square}(A^3 + 2AB) - 3(\square + 2B)$[/tex]

Step 2: [tex][tex]$(A^3 + 2B)\left(A - \square\right)$[/tex][/tex]



Answer :

Given the polynomial: [tex]\(A^4+2 A^3 B-3 A-6 B\)[/tex]

Step 1: Factor by grouping.

Let's break the polynomial into two groups:
[tex]\[ A^4 + 2A^3B \][/tex] and [tex]\[-3A - 6B \][/tex]

Now, factor each group separately:

1. In the first group [tex]\( A^4 + 2A^3B \)[/tex]:
[tex]\[ A^4 + 2A^3B = A^3(A + 2B) \][/tex]

2. In the second group [tex]\(-3A - 6B\)[/tex]:
[tex]\[ -3A - 6B = -3(A + 2B) \][/tex]

Combining these factorizations gives:
[tex]\[ A^3(A + 2B) - 3(A + 2B) \][/tex]

Step 2: Factor out the common term [tex]\((A + 2B)\)[/tex]:
[tex]\[ (A + 2B)(A^3 - 3) \][/tex]

So, the filled-in blanks should be:

Step 1: [tex]\(A^{\mathbf{3}}(A+2 B)-3(\mathbf{A}+2 B)\)[/tex]

Step 2: [tex]\((A+2 \mathbf{B})\left(A^3-\mathbf{3}\right)\)[/tex]

The completed expression is:
[tex]\[ (A+2B)(A^3-3) \][/tex]