Question
Write the polynomial as a product of linear factors \( x^{3}+5 x^{2}-9 x-45 \) \( (x-[?])(x+3)(x+[\quad]) \)
Ask by Salazar Warren. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The polynomial \( x^{3}+5 x^{2}-9 x-45 \) can be factored into \( (x-3)(x+3)(x+5) \).
Solution
Factor the expression by following steps:
- step0: Factor:
\(x^{3}+5x^{2}-9x-45\)
- step1: Rewrite the expression:
\(x^{2}\times x+x^{2}\times 5-9x-9\times 5\)
- step2: Factor the expression:
\(x^{2}\left(x+5\right)-9\left(x+5\right)\)
- step3: Factor the expression:
\(\left(x^{2}-9\right)\left(x+5\right)\)
- step4: Factor the expression:
\(\left(x-3\right)\left(x+3\right)\left(x+5\right)\)
To express the polynomial \( x^{3}+5 x^{2}-9 x-45 \) as a product of linear factors, we can factor it as follows:
\[
x^{3}+5 x^{2}-9 x-45 = (x-3)(x+3)(x+5)
\]
Thus, the polynomial can be written in the form:
\[
(x-3)(x+3)(x+5)
\]
So, the missing values in your expression are:
- The first factor is \( x-3 \)
- The second factor is \( x+3 \)
- The third factor is \( x+5 \)
Therefore, the complete expression is:
\[
(x-3)(x+3)(x+5)
\]
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Simplify this solution Bonus Knowledge
To factor the polynomial \( x^{3}+5 x^{2}-9 x-45 \) into linear factors, we first look for rational roots using the Rational Root Theorem. We test \( x = -3 \) and find it indeed is a root. Dividing the polynomial by \( x + 3 \) yields \( (x + 3)(x^2 + 2x - 15) \). Further factoring \( x^2 + 2x - 15 \) gives \( (x + 5)(x - 3) \). Thus, the polynomial can be written as \( (x + 3)(x + 5)(x - 3) \). So, the full expression is \( (x - 3)(x + 3)(x + 5) \).
