Answer :
To factor the polynomial [tex]\(x^3 + 5x^2 + 9x + 45\)[/tex] completely, we can follow these steps:
1. Look for a Common Factor: First, check if there's a greatest common factor (GCF) for all the terms of the polynomial. In this case, there are no common factors other than 1.
2. Group the Terms: Since the polynomial is a cubic (third-degree) polynomial, one effective strategy is to group the terms to factor by grouping:
- Group the terms as [tex]\((x^3 + 5x^2)\)[/tex] and [tex]\((9x + 45)\)[/tex].
3. Factor Each Group:
- In the first group, [tex]\(x^3 + 5x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex], giving us:
[tex]\[
x^2(x + 5)
\][/tex]
- In the second group, [tex]\(9x + 45\)[/tex], we can factor out 9, giving us:
[tex]\[
9(x + 5)
\][/tex]
4. Combine the Groups: Notice that both terms now share a common factor of [tex]\((x + 5)\)[/tex]:
[tex]\[
x^2(x + 5) + 9(x + 5) = (x + 5)(x^2 + 9)
\][/tex]
5. Check for Further Factoring: The term [tex]\((x^2 + 9)\)[/tex] does not factor further into real numbers since it does not have real roots. Therefore, the polynomial is factored completely over the real numbers as:
[tex]\[
(x + 5)(x^2 + 9)
\][/tex]
This is the complete factorization of the polynomial [tex]\(x^3 + 5x^2 + 9x + 45\)[/tex].
1. Look for a Common Factor: First, check if there's a greatest common factor (GCF) for all the terms of the polynomial. In this case, there are no common factors other than 1.
2. Group the Terms: Since the polynomial is a cubic (third-degree) polynomial, one effective strategy is to group the terms to factor by grouping:
- Group the terms as [tex]\((x^3 + 5x^2)\)[/tex] and [tex]\((9x + 45)\)[/tex].
3. Factor Each Group:
- In the first group, [tex]\(x^3 + 5x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex], giving us:
[tex]\[
x^2(x + 5)
\][/tex]
- In the second group, [tex]\(9x + 45\)[/tex], we can factor out 9, giving us:
[tex]\[
9(x + 5)
\][/tex]
4. Combine the Groups: Notice that both terms now share a common factor of [tex]\((x + 5)\)[/tex]:
[tex]\[
x^2(x + 5) + 9(x + 5) = (x + 5)(x^2 + 9)
\][/tex]
5. Check for Further Factoring: The term [tex]\((x^2 + 9)\)[/tex] does not factor further into real numbers since it does not have real roots. Therefore, the polynomial is factored completely over the real numbers as:
[tex]\[
(x + 5)(x^2 + 9)
\][/tex]
This is the complete factorization of the polynomial [tex]\(x^3 + 5x^2 + 9x + 45\)[/tex].
