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Fill in the blanks to complete the factoring process.

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


Sagot :

Sure, let's complete the factoring process for the polynomial [tex]\( A^4 + 2A^3B - 3A - 6B \)[/tex] step-by-step:

1. Identify possible groupings: Start by examining the polynomial [tex]\( A^4 + 2A^3B - 3A - 6B \)[/tex] and look for possible groupings of terms that could facilitate factoring:

Group terms as:
[tex]\[ (A^4 + 2A^3B) + (-3A - 6B) \][/tex]

2. Factor out common factors from each group:

- From the first group, [tex]\( A^4 + 2A^3B \)[/tex]:
We can factor out [tex]\( A^3 \)[/tex]:
[tex]\[ A^3(A + 2B) \][/tex]

- From the second group, [tex]\( -3A - 6B \)[/tex]:
We can factor out [tex]\( -3 \)[/tex]:
[tex]\[ -3(A + 2B) \][/tex]

3. Combine the factored groups:

Now we can see a common factor [tex]\( (A + 2B) \)[/tex] in both terms:
[tex]\[ A^3(A + 2B) - 3(A + 2B) \][/tex]

4. Factor out the common binomial:

Since [tex]\( (A + 2B) \)[/tex] is common in both terms, we can factor it out:
[tex]\[ (A + 2B)(A^3 - 3) \][/tex]

Therefore, the factored form of the polynomial [tex]\( A^4 + 2A^3B - 3A - 6B \)[/tex] is:
[tex]\[ (A + 2B)(A^3 - 3) \][/tex]