Answer :
Sure! Let's go through the process of factoring these expressions using the Greatest Common Factor (GCF).
### 11) [tex]\( 20 - 35n^2 - 20n^3 \)[/tex]
1. Identify the GCF of the coefficients: 20, 35, and 20. The GCF is 5.
2. Identify the variable terms. Here, the smallest power of [tex]\( n \)[/tex] is 0 (as [tex]\( 20n^0 \)[/tex] counts as having no [tex]\( n \)[/tex]).
3. Factor out 5:
[tex]\[
5(4n^3 + 7n^2 - 4)
\][/tex]
### 12) [tex]\( 9x^6 - 63x^3 - 90x^2 \)[/tex]
1. Identify the GCF of the coefficients: 9, 63, and 90. The GCF is 9.
2. The smallest power of [tex]\( x \)[/tex] common in all terms is [tex]\( x^2 \)[/tex].
3. Factor out [tex]\( 9x^2 \)[/tex]:
[tex]\[
9x^2(x^4 - 7x - 10)
\][/tex]
### 13) [tex]\( -3k^3 + 15k^2 - 6k \)[/tex]
1. The GCF of the coefficients [tex]\(-3\)[/tex], 15, and [tex]\(-6\)[/tex] is 3.
2. The smallest power of [tex]\( k \)[/tex] is 1.
3. Factor out [tex]\(-3k\)[/tex]:
[tex]\[
-3k(k^2 - 5k + 2)
\][/tex]
### 14) [tex]\( 50p^3 + 50p^2 - 20 \)[/tex]
1. The GCF of the coefficients 50, 50, and [tex]\(-20\)[/tex] is 10.
2. There are no common [tex]\( p \)[/tex] terms in all coefficients.
3. Factor out 10:
[tex]\[
10(5p^3 + 5p^2 - 2)
\][/tex]
### 15) [tex]\( 32n^3 + 28n - 20 \)[/tex]
1. The GCF of the coefficients 32, 28, and [tex]\(-20\)[/tex] is 4.
2. There are no common [tex]\( n \)[/tex] terms in all coefficients.
3. Factor out 4:
[tex]\[
4(8n^3 + 7n - 5)
\][/tex]
### 16) [tex]\( -90x^5 + 100x + 60 \)[/tex]
1. The GCF of the coefficients [tex]\(-90\)[/tex], 100, and 60 is 10.
2. Since [tex]\( x \)[/tex] isn't common across all terms, just factor out numbers.
3. Factor out [tex]\(-10\)[/tex]:
[tex]\[
-10(9x^5 - 10x - 6)
\][/tex]
### 17) [tex]\( 3m^2 + 9m + 27 \)[/tex]
1. The GCF of the coefficients 3, 9, and 27 is 3.
2. There are no common [tex]\( m \)[/tex] factors in all terms.
3. Factor out 3:
[tex]\[
3(m^2 + 3m + 9)
\][/tex]
### 18) [tex]\( 12r^2 + 4r - 12 \)[/tex]
1. The GCF of the coefficients 12, 4, and [tex]\(-12\)[/tex] is 4.
2. There are no common [tex]\( r \)[/tex] factors in all terms.
3. Factor out 4:
[tex]\[
4(3r^2 + r - 3)
\][/tex]
### 19) [tex]\( 64 + 40x^2 + 72x \)[/tex]
1. The GCF of the coefficients 64, 40, and 72 is 8.
2. There isn't a common [tex]\( x \)[/tex] in all terms.
3. Factor out 8:
[tex]\[
8(5x^2 + 9x + 8)
\][/tex]
### 20) [tex]\( -18n^2 + 15n - 15 \)[/tex]
1. The GCF of the coefficients [tex]\(-18\)[/tex], 15, and [tex]\(-15\)[/tex] is 3.
2. There is no shared [tex]\( n \)[/tex] across all terms.
3. Factor out [tex]\(-3\)[/tex]:
[tex]\[
-3(6n^2 - 5n + 5)
\][/tex]
By identifying and factoring out the GCF, we can simplify these expressions effectively. If you have any more questions on factoring, feel free to ask!
### 11) [tex]\( 20 - 35n^2 - 20n^3 \)[/tex]
1. Identify the GCF of the coefficients: 20, 35, and 20. The GCF is 5.
2. Identify the variable terms. Here, the smallest power of [tex]\( n \)[/tex] is 0 (as [tex]\( 20n^0 \)[/tex] counts as having no [tex]\( n \)[/tex]).
3. Factor out 5:
[tex]\[
5(4n^3 + 7n^2 - 4)
\][/tex]
### 12) [tex]\( 9x^6 - 63x^3 - 90x^2 \)[/tex]
1. Identify the GCF of the coefficients: 9, 63, and 90. The GCF is 9.
2. The smallest power of [tex]\( x \)[/tex] common in all terms is [tex]\( x^2 \)[/tex].
3. Factor out [tex]\( 9x^2 \)[/tex]:
[tex]\[
9x^2(x^4 - 7x - 10)
\][/tex]
### 13) [tex]\( -3k^3 + 15k^2 - 6k \)[/tex]
1. The GCF of the coefficients [tex]\(-3\)[/tex], 15, and [tex]\(-6\)[/tex] is 3.
2. The smallest power of [tex]\( k \)[/tex] is 1.
3. Factor out [tex]\(-3k\)[/tex]:
[tex]\[
-3k(k^2 - 5k + 2)
\][/tex]
### 14) [tex]\( 50p^3 + 50p^2 - 20 \)[/tex]
1. The GCF of the coefficients 50, 50, and [tex]\(-20\)[/tex] is 10.
2. There are no common [tex]\( p \)[/tex] terms in all coefficients.
3. Factor out 10:
[tex]\[
10(5p^3 + 5p^2 - 2)
\][/tex]
### 15) [tex]\( 32n^3 + 28n - 20 \)[/tex]
1. The GCF of the coefficients 32, 28, and [tex]\(-20\)[/tex] is 4.
2. There are no common [tex]\( n \)[/tex] terms in all coefficients.
3. Factor out 4:
[tex]\[
4(8n^3 + 7n - 5)
\][/tex]
### 16) [tex]\( -90x^5 + 100x + 60 \)[/tex]
1. The GCF of the coefficients [tex]\(-90\)[/tex], 100, and 60 is 10.
2. Since [tex]\( x \)[/tex] isn't common across all terms, just factor out numbers.
3. Factor out [tex]\(-10\)[/tex]:
[tex]\[
-10(9x^5 - 10x - 6)
\][/tex]
### 17) [tex]\( 3m^2 + 9m + 27 \)[/tex]
1. The GCF of the coefficients 3, 9, and 27 is 3.
2. There are no common [tex]\( m \)[/tex] factors in all terms.
3. Factor out 3:
[tex]\[
3(m^2 + 3m + 9)
\][/tex]
### 18) [tex]\( 12r^2 + 4r - 12 \)[/tex]
1. The GCF of the coefficients 12, 4, and [tex]\(-12\)[/tex] is 4.
2. There are no common [tex]\( r \)[/tex] factors in all terms.
3. Factor out 4:
[tex]\[
4(3r^2 + r - 3)
\][/tex]
### 19) [tex]\( 64 + 40x^2 + 72x \)[/tex]
1. The GCF of the coefficients 64, 40, and 72 is 8.
2. There isn't a common [tex]\( x \)[/tex] in all terms.
3. Factor out 8:
[tex]\[
8(5x^2 + 9x + 8)
\][/tex]
### 20) [tex]\( -18n^2 + 15n - 15 \)[/tex]
1. The GCF of the coefficients [tex]\(-18\)[/tex], 15, and [tex]\(-15\)[/tex] is 3.
2. There is no shared [tex]\( n \)[/tex] across all terms.
3. Factor out [tex]\(-3\)[/tex]:
[tex]\[
-3(6n^2 - 5n + 5)
\][/tex]
By identifying and factoring out the GCF, we can simplify these expressions effectively. If you have any more questions on factoring, feel free to ask!
