Question
or
Factor completely.
5x^3+10x^2+9x+18
Answers
Answered by
GPT-4o mini
To factor the polynomial \( 5x^3 + 10x^2 + 9x + 18 \) completely, we can first group the terms:
\[
(5x^3 + 10x^2) + (9x + 18)
\]
Now, factor out the common factors from each group:
\[
5x^2(x + 2) + 9(x + 2)
\]
Next, we can factor out the common binomial factor \( (x + 2) \):
\[
(5x^2 + 9)(x + 2)
\]
Now, we need to check if \( 5x^2 + 9 \) can be factored further. Since it is a sum of squares (it does not have real roots), we leave it as it is.
Thus, the completely factored form of the polynomial \( 5x^3 + 10x^2 + 9x + 18 \) is:
\[
(5x^2 + 9)(x + 2)
\]
\[
(5x^3 + 10x^2) + (9x + 18)
\]
Now, factor out the common factors from each group:
\[
5x^2(x + 2) + 9(x + 2)
\]
Next, we can factor out the common binomial factor \( (x + 2) \):
\[
(5x^2 + 9)(x + 2)
\]
Now, we need to check if \( 5x^2 + 9 \) can be factored further. Since it is a sum of squares (it does not have real roots), we leave it as it is.
Thus, the completely factored form of the polynomial \( 5x^3 + 10x^2 + 9x + 18 \) is:
\[
(5x^2 + 9)(x + 2)
\]