Sure, let's go through the factoring process for the given polynomial [tex]\( A^4 + 2 A^3 B - 3 A - 6 B \)[/tex] step-by-step.
Given polynomial:
[tex]\[ A^4 + 2 A^3 B - 3 A - 6 B \][/tex]
We aim to factor the polynomial. Let's rewrite and look for common factors:
[tex]\[ A^0(A+2 B) - 3(0 + 2 B) \][/tex]
This step seems incorrect as it does not align with an appropriate factoring technique for the polynomial. Let's start over:
Given polynomial:
[tex]\[ A^4 + 2 A^3 B - 3 A - 6 B \][/tex]
Step 1: Group the polynomial into two parts:
[tex]\[ (A^4 + 2 A^3 B) + (-3 A - 6 B) \][/tex]
Step 2: Factor out the greatest common factor from each group.
For the first group [tex]\( A^4 + 2 A^3 B \)[/tex]:
The common factor is [tex]\( A^3 \)[/tex]:
[tex]\[ A^3 (A + 2 B) \][/tex]
For the second group [tex]\( -3 A - 6 B \)[/tex]:
The common factor is [tex]\( -3 \)[/tex]:
[tex]\[ -3 (A + 2 B) \][/tex]
Step 3: Write the polynomial as a product of common factors:
[tex]\[ A^3(A + 2 B) - 3(A + 2 B) \][/tex]
Step 4: Factor out the common binomial [tex]\( A + 2 B \)[/tex]:
[tex]\[ (A + 2 B)(A^3 - 3) \][/tex]
Now we have factored the polynomial:
[tex]\[ (A + 2 B)(A^3 - 3) \][/tex]
Thus, the complete factorization of the polynomial [tex]\( A^4 + 2 A^3 B - 3 A - 6 B \)[/tex] is:
[tex]\[ (A + 2 B)(A^3 - 3) \][/tex]
