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Answer :
To factor the polynomial [tex]\(x^3 + 5x^2 + 9x + 45\)[/tex] completely, we can follow these steps:
1. Check for a common factor:
Look for any common factor in all the terms. In this case, there is no common factor besides 1, so we proceed to the next step.
2. Factor by grouping:
Group the terms in pairs to see if we can factor them more easily:
[tex]\[
(x^3 + 5x^2) + (9x + 45)
\][/tex]
3. Factor each group separately:
- For the first group [tex]\(x^3 + 5x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x + 5)
\][/tex]
- For the second group [tex]\(9x + 45\)[/tex], we can factor out 9:
[tex]\[
9(x + 5)
\][/tex]
4. Factor the expression:
Now, notice that both groups contain the common factor [tex]\((x + 5)\)[/tex]. We can factor that out:
[tex]\[
(x^2 + 9)(x + 5)
\][/tex]
5. Final check:
Ensure that each factor is fully simplified. In this case, [tex]\((x^2 + 9)\)[/tex] cannot be factored further using real numbers because it does not have real roots.
So, the completely factored form of the polynomial [tex]\(x^3 + 5x^2 + 9x + 45\)[/tex] is:
[tex]\[
(x + 5)(x^2 + 9)
\][/tex]
That's the completely factored form of the given polynomial!
1. Check for a common factor:
Look for any common factor in all the terms. In this case, there is no common factor besides 1, so we proceed to the next step.
2. Factor by grouping:
Group the terms in pairs to see if we can factor them more easily:
[tex]\[
(x^3 + 5x^2) + (9x + 45)
\][/tex]
3. Factor each group separately:
- For the first group [tex]\(x^3 + 5x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x + 5)
\][/tex]
- For the second group [tex]\(9x + 45\)[/tex], we can factor out 9:
[tex]\[
9(x + 5)
\][/tex]
4. Factor the expression:
Now, notice that both groups contain the common factor [tex]\((x + 5)\)[/tex]. We can factor that out:
[tex]\[
(x^2 + 9)(x + 5)
\][/tex]
5. Final check:
Ensure that each factor is fully simplified. In this case, [tex]\((x^2 + 9)\)[/tex] cannot be factored further using real numbers because it does not have real roots.
So, the completely factored form of the polynomial [tex]\(x^3 + 5x^2 + 9x + 45\)[/tex] is:
[tex]\[
(x + 5)(x^2 + 9)
\][/tex]
That's the completely factored form of the given polynomial!
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