Answer :
Sure! Let's factor the polynomial [tex]\(45x^3 - 63x^2 + 20x - 28\)[/tex] by grouping.
1. Group the terms:
Split the expression into two groups:
[tex]\[
(45x^3 - 63x^2) + (20x - 28)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- For the first group [tex]\(45x^3 - 63x^2\)[/tex], the GCF is [tex]\(9x^2\)[/tex]:
[tex]\[
45x^3 - 63x^2 = 9x^2(5x - 7)
\][/tex]
- For the second group [tex]\(20x - 28\)[/tex], the GCF is [tex]\(4\)[/tex]:
[tex]\[
20x - 28 = 4(5x - 7)
\][/tex]
3. Rewrite the expression with factored groups:
Now, the expression looks like this:
[tex]\[
9x^2(5x - 7) + 4(5x - 7)
\][/tex]
4. Factor out the common binomial factor:
Notice that both terms have a common binomial factor of [tex]\((5x - 7)\)[/tex]:
[tex]\[
(9x^2 + 4)(5x - 7)
\][/tex]
So, the factored form of the polynomial [tex]\(45x^3 - 63x^2 + 20x - 28\)[/tex] by grouping is [tex]\((9x^2 + 4)(5x - 7)\)[/tex].
1. Group the terms:
Split the expression into two groups:
[tex]\[
(45x^3 - 63x^2) + (20x - 28)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- For the first group [tex]\(45x^3 - 63x^2\)[/tex], the GCF is [tex]\(9x^2\)[/tex]:
[tex]\[
45x^3 - 63x^2 = 9x^2(5x - 7)
\][/tex]
- For the second group [tex]\(20x - 28\)[/tex], the GCF is [tex]\(4\)[/tex]:
[tex]\[
20x - 28 = 4(5x - 7)
\][/tex]
3. Rewrite the expression with factored groups:
Now, the expression looks like this:
[tex]\[
9x^2(5x - 7) + 4(5x - 7)
\][/tex]
4. Factor out the common binomial factor:
Notice that both terms have a common binomial factor of [tex]\((5x - 7)\)[/tex]:
[tex]\[
(9x^2 + 4)(5x - 7)
\][/tex]
So, the factored form of the polynomial [tex]\(45x^3 - 63x^2 + 20x - 28\)[/tex] by grouping is [tex]\((9x^2 + 4)(5x - 7)\)[/tex].
