Sure! Let's factor each of the given polynomials step-by-step, identifying and extracting the greatest common monomial factor in each case.
7. [tex]\(4x^3 + 4x^2 + 4x\)[/tex]
First, identify the greatest common factor (GCF) in each term:
- Each term includes a factor of [tex]\(4x\)[/tex].
Therefore, we can factor out [tex]\(4x\)[/tex]:
[tex]\[ 4x^3 + 4x^2 + 4x = 4x(x^2 + x + 1) \][/tex]
So, the factored form is:
[tex]\[ \boxed{4x(x^2 + x + 1)} \][/tex]
8. [tex]\(5x^2 - 10x\)[/tex]
First, identify the GCF in each term:
- Each term includes a factor of [tex]\(5x\)[/tex].
Therefore, we can factor out [tex]\(5x\)[/tex]:
[tex]\[ 5x^2 - 10x = 5x(x - 2) \][/tex]
So, the factored form is:
[tex]\[ \boxed{5x(x - 2)} \][/tex]
9. [tex]\(4a^4 + 8a^3 + 12a^2\)[/tex]
First, identify the GCF in each term:
- Each term includes a factor of [tex]\(4a^2\)[/tex].
Therefore, we can factor out [tex]\(4a^2\)[/tex]:
[tex]\[ 4a^4 + 8a^3 + 12a^2 = 4a^2(a^2 + 2a + 3) \][/tex]
So, the factored form is:
[tex]\[ \boxed{4a^2(a^2 + 2a + 3)} \][/tex]
10. [tex]\(-10a^2 bc - 4ab^2 c - 16abc\)[/tex]
First, identify the GCF in each term:
- Each term includes a factor of [tex]\(-2abc\)[/tex].
Therefore, we can factor out [tex]\(-2abc\)[/tex]:
[tex]\[ -10a^2 bc - 4ab^2 c - 16abc = -2abc(5a + 2b + 8) \][/tex]
So, the factored form is:
[tex]\[ \boxed{-2abc(5a + 2b + 8)} \][/tex]
11. [tex]\(40aa^2 + 60ab^2\)[/tex]
First, identify the GCF in each term:
- Each term includes a factor of [tex]\(20a\)[/tex].
Therefore, we can factor out [tex]\(20a\)[/tex]:
[tex]\[ 40aa^2 + 60ab^2 = 20a(2a^2 + 3b^2) \][/tex]
So, the factored form is:
[tex]\[ \boxed{20a(2a^2 + 3b^2)} \][/tex]
12. [tex]\(18xy - 9xy^2 + 36x^2y\)[/tex]
First, identify the GCF in each term:
- Each term includes a factor of [tex]\(9xy\)[/tex].
Therefore, we can factor out [tex]\(9xy\)[/tex]:
[tex]\[ 18xy - 9xy^2 + 36x^2y = 9xy(2 - y + 4x) \][/tex]
So, the factored form is:
[tex]\[ \boxed{9xy(2 - y + 4x)} \][/tex]
All the steps involve identifying the common monomial factor in each polynomial and factoring it out, simplifying the expression into a product of that common factor and the remaining polynomial. This results in the factored forms provided.
