Answer :
Sure! Let's break down each problem to find the Greatest Common Factor (GCF) and factor the expressions. Remember, factoring involves expressing an expression as a product of its factors. Here, we will focus on factoring out the GCF for each expression.
### Problem 11: [tex]\(20 - 35n^2 - 20n^3\)[/tex]
1. Identify the GCF: The coefficients are 20, 35, and 20. The GCF of these numbers is 5.
2. Factor out the GCF:
[tex]\[
20 - 35n^2 - 20n^3 = -5(4n^3 + 7n^2 - 4)
\][/tex]
### Problem 12: [tex]\(9x^6 - 63x^3 - 90x^2\)[/tex]
1. Identify the GCF: The coefficients are 9, 63, and 90, with a GCF of 9. The variable factor common to all terms is [tex]\(x^2\)[/tex].
2. Factor out the GCF:
[tex]\[
9x^6 - 63x^3 - 90x^2 = 9x^2(x^4 - 7x - 10)
\][/tex]
### Problem 13: [tex]\(-3k^3 + 15k^2 - 6k\)[/tex]
1. Identify the GCF: The coefficients are 3, 15, and 6, with a GCF of 3. The variable factor common to all terms is [tex]\(k\)[/tex].
2. Factor out the GCF:
[tex]\[
-3k^3 + 15k^2 - 6k = -3k(k^2 - 5k + 2)
\][/tex]
### Problem 14: [tex]\(50p^3 + 50p^2 - 20\)[/tex]
1. Identify the GCF: The coefficients 50, 50, and 20 have a GCF of 10.
2. Factor out the GCF:
[tex]\[
50p^3 + 50p^2 - 20 = 10(5p^3 + 5p^2 - 2)
\][/tex]
### Problem 15: [tex]\(32n^3 + 28n - 20\)[/tex]
1. Identify the GCF: The coefficients 32, 28, and 20 have a GCF of 4.
2. Factor out the GCF:
[tex]\[
32n^3 + 28n - 20 = 4(8n^3 + 7n - 5)
\][/tex]
### Problem 16: [tex]\(-90x^5 + 100x + 60\)[/tex]
1. Identify the GCF: The coefficients 90, 100, and 60 have a GCF of 10.
2. Factor out the GCF:
[tex]\[
-90x^5 + 100x + 60 = -10(9x^5 - 10x - 6)
\][/tex]
### Problem 17: [tex]\(3m^2 + 9m + 27\)[/tex]
1. Identify the GCF: The coefficients 3, 9, and 27 have a GCF of 3.
2. Factor out the GCF:
[tex]\[
3m^2 + 9m + 27 = 3(m^2 + 3m + 9)
\][/tex]
### Problem 18: [tex]\(12r^2 + 4r - 12\)[/tex]
1. Identify the GCF: The coefficients 12, 4, and 12 have a GCF of 4.
2. Factor out the GCF:
[tex]\[
12r^2 + 4r - 12 = 4(3r^2 + r - 3)
\][/tex]
### Problem 19: [tex]\(64 + 40x^2 + 72x\)[/tex]
1. Identify the GCF: The coefficients 64, 40, and 72 have a GCF of 8.
2. Factor out the GCF:
[tex]\[
64 + 40x^2 + 72x = 8(5x^2 + 9x + 8)
\][/tex]
### Problem 20: [tex]\(-18n^2 + 15n - 15\)[/tex]
1. Identify the GCF: The coefficients 18, 15, and 15 have a GCF of 3.
2. Factor out the GCF:
[tex]\[
-18n^2 + 15n - 15 = -3(6n^2 - 5n + 5)
\][/tex]
Remember, in each step we first find the GCF of all the terms, and then factor it out from the expression to simplify it. Each expression then becomes a product of its GCF and the remaining polynomial.
### Problem 11: [tex]\(20 - 35n^2 - 20n^3\)[/tex]
1. Identify the GCF: The coefficients are 20, 35, and 20. The GCF of these numbers is 5.
2. Factor out the GCF:
[tex]\[
20 - 35n^2 - 20n^3 = -5(4n^3 + 7n^2 - 4)
\][/tex]
### Problem 12: [tex]\(9x^6 - 63x^3 - 90x^2\)[/tex]
1. Identify the GCF: The coefficients are 9, 63, and 90, with a GCF of 9. The variable factor common to all terms is [tex]\(x^2\)[/tex].
2. Factor out the GCF:
[tex]\[
9x^6 - 63x^3 - 90x^2 = 9x^2(x^4 - 7x - 10)
\][/tex]
### Problem 13: [tex]\(-3k^3 + 15k^2 - 6k\)[/tex]
1. Identify the GCF: The coefficients are 3, 15, and 6, with a GCF of 3. The variable factor common to all terms is [tex]\(k\)[/tex].
2. Factor out the GCF:
[tex]\[
-3k^3 + 15k^2 - 6k = -3k(k^2 - 5k + 2)
\][/tex]
### Problem 14: [tex]\(50p^3 + 50p^2 - 20\)[/tex]
1. Identify the GCF: The coefficients 50, 50, and 20 have a GCF of 10.
2. Factor out the GCF:
[tex]\[
50p^3 + 50p^2 - 20 = 10(5p^3 + 5p^2 - 2)
\][/tex]
### Problem 15: [tex]\(32n^3 + 28n - 20\)[/tex]
1. Identify the GCF: The coefficients 32, 28, and 20 have a GCF of 4.
2. Factor out the GCF:
[tex]\[
32n^3 + 28n - 20 = 4(8n^3 + 7n - 5)
\][/tex]
### Problem 16: [tex]\(-90x^5 + 100x + 60\)[/tex]
1. Identify the GCF: The coefficients 90, 100, and 60 have a GCF of 10.
2. Factor out the GCF:
[tex]\[
-90x^5 + 100x + 60 = -10(9x^5 - 10x - 6)
\][/tex]
### Problem 17: [tex]\(3m^2 + 9m + 27\)[/tex]
1. Identify the GCF: The coefficients 3, 9, and 27 have a GCF of 3.
2. Factor out the GCF:
[tex]\[
3m^2 + 9m + 27 = 3(m^2 + 3m + 9)
\][/tex]
### Problem 18: [tex]\(12r^2 + 4r - 12\)[/tex]
1. Identify the GCF: The coefficients 12, 4, and 12 have a GCF of 4.
2. Factor out the GCF:
[tex]\[
12r^2 + 4r - 12 = 4(3r^2 + r - 3)
\][/tex]
### Problem 19: [tex]\(64 + 40x^2 + 72x\)[/tex]
1. Identify the GCF: The coefficients 64, 40, and 72 have a GCF of 8.
2. Factor out the GCF:
[tex]\[
64 + 40x^2 + 72x = 8(5x^2 + 9x + 8)
\][/tex]
### Problem 20: [tex]\(-18n^2 + 15n - 15\)[/tex]
1. Identify the GCF: The coefficients 18, 15, and 15 have a GCF of 3.
2. Factor out the GCF:
[tex]\[
-18n^2 + 15n - 15 = -3(6n^2 - 5n + 5)
\][/tex]
Remember, in each step we first find the GCF of all the terms, and then factor it out from the expression to simplify it. Each expression then becomes a product of its GCF and the remaining polynomial.
