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Answer :
Sure! Let's factor the polynomial [tex]\(2x^3 + 5x^2 - 18x - 45\)[/tex] by grouping, step by step.
1. Group the terms:
- The polynomial [tex]\(2x^3 + 5x^2 - 18x - 45\)[/tex] can be divided into two groups: [tex]\((2x^3 + 5x^2)\)[/tex] and [tex]\((-18x - 45)\)[/tex].
2. Factor out the greatest common factor in each group:
- In the first group [tex]\((2x^3 + 5x^2)\)[/tex], the greatest common factor is [tex]\(x^2\)[/tex]. So, we factor it out:
[tex]\[
x^2(2x + 5)
\][/tex]
- In the second group [tex]\((-18x - 45)\)[/tex], the greatest common factor is [tex]\(-9\)[/tex]. So, we factor it out:
[tex]\[
-9(2x + 5)
\][/tex]
3. Recognize and combine the common factor:
- We notice that [tex]\((2x + 5)\)[/tex] is a common factor in both groups. So, we can combine the groups:
[tex]\[
(x^2 - 9)(2x + 5)
\][/tex]
4. Factor completely if possible:
- The expression [tex]\((x^2 - 9)\)[/tex] is a difference of squares, which can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
5. Final factored form:
- Substitute [tex]\((x - 3)(x + 3)\)[/tex] back into the expression:
[tex]\[
(x - 3)(x + 3)(2x + 5)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(2x^3 + 5x^2 - 18x - 45\)[/tex] is [tex]\((x - 3)(x + 3)(2x + 5)\)[/tex].
1. Group the terms:
- The polynomial [tex]\(2x^3 + 5x^2 - 18x - 45\)[/tex] can be divided into two groups: [tex]\((2x^3 + 5x^2)\)[/tex] and [tex]\((-18x - 45)\)[/tex].
2. Factor out the greatest common factor in each group:
- In the first group [tex]\((2x^3 + 5x^2)\)[/tex], the greatest common factor is [tex]\(x^2\)[/tex]. So, we factor it out:
[tex]\[
x^2(2x + 5)
\][/tex]
- In the second group [tex]\((-18x - 45)\)[/tex], the greatest common factor is [tex]\(-9\)[/tex]. So, we factor it out:
[tex]\[
-9(2x + 5)
\][/tex]
3. Recognize and combine the common factor:
- We notice that [tex]\((2x + 5)\)[/tex] is a common factor in both groups. So, we can combine the groups:
[tex]\[
(x^2 - 9)(2x + 5)
\][/tex]
4. Factor completely if possible:
- The expression [tex]\((x^2 - 9)\)[/tex] is a difference of squares, which can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
5. Final factored form:
- Substitute [tex]\((x - 3)(x + 3)\)[/tex] back into the expression:
[tex]\[
(x - 3)(x + 3)(2x + 5)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(2x^3 + 5x^2 - 18x - 45\)[/tex] is [tex]\((x - 3)(x + 3)(2x + 5)\)[/tex].
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