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Answer :
To factor the polynomial [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex] completely, we can use a method called grouping, which involves rearranging and grouping terms to help us find common factors. Let's go through this step-by-step:
1. Group the terms:
The polynomial is [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex]. We'll divide it into two groups:
[tex]\[
(8x^3 + 20x^2) + (-18x - 45)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group [tex]\(8x^3 + 20x^2\)[/tex], the GCF is [tex]\(4x^2\)[/tex].
- In the second group [tex]\(-18x - 45\)[/tex], the GCF is [tex]\(-3\)[/tex].
Rewrite each group:
[tex]\[
4x^2 (2x + 5) - 3(2x + 5)
\][/tex]
3. Factor by grouping:
Notice that both terms now include the common factor [tex]\((2x + 5)\)[/tex]. So, we can factor that out:
[tex]\[
(4x^2 - 3)(2x + 5)
\][/tex]
4. Check the factorization:
- The factor [tex]\(4x^2 - 3\)[/tex] does not factor further because it is a simple quadratic with no rational roots.
- The factor [tex]\(2x + 5\)[/tex] is a linear binomial and is already in its simplest form.
Therefore, the completely factored form of [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex] is:
[tex]\[
(4x^2 - 3)(2x + 5)
\][/tex]
1. Group the terms:
The polynomial is [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex]. We'll divide it into two groups:
[tex]\[
(8x^3 + 20x^2) + (-18x - 45)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group [tex]\(8x^3 + 20x^2\)[/tex], the GCF is [tex]\(4x^2\)[/tex].
- In the second group [tex]\(-18x - 45\)[/tex], the GCF is [tex]\(-3\)[/tex].
Rewrite each group:
[tex]\[
4x^2 (2x + 5) - 3(2x + 5)
\][/tex]
3. Factor by grouping:
Notice that both terms now include the common factor [tex]\((2x + 5)\)[/tex]. So, we can factor that out:
[tex]\[
(4x^2 - 3)(2x + 5)
\][/tex]
4. Check the factorization:
- The factor [tex]\(4x^2 - 3\)[/tex] does not factor further because it is a simple quadratic with no rational roots.
- The factor [tex]\(2x + 5\)[/tex] is a linear binomial and is already in its simplest form.
Therefore, the completely factored form of [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex] is:
[tex]\[
(4x^2 - 3)(2x + 5)
\][/tex]
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