Problem 71 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-9 x-18$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x^{3}+2x^{2}-9x-18\) is \((x+2) (x+3) (x-3)\).

Step 1: Test for Factorability

To test for factorability, we need to find a value of \(x\) that makes the expression equal to zero. By trial and error (guessing), or using rational root theorem we find that \(x=-2\), \(x=-3\) and \(x=3\) are roots of the equation, meaning they make the expression equal to zero. So, we can conclude that the polynomial is not prime since it has rational roots.

Step 2: Factor the polynomial

Since we've identified that \(x=-2\), \(x=-3\) and \(x=3\) are solutions for the equation, un-multiply or factor the polynomial to find the component polynomials. This gives a set of first degree polynomials (factors) as \( (x+2), (x+3), (x-3)\).

Step 3: Write out the complete factorization

Once you've identified the correct factors, write the polynomial as a product of these factors. Therefore, \(x^{3}+2x^{2}-9x-18 = (x+2) (x+3) (x-3)\). This is the completely factored form of the given polynomial.