Problem 72 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$x^{3}+3 x^{2}-25 x-75$$

Step-by-Step Solution

Answer

So, the complete factorization of the original polynomial \(x^{3}+3 x^{2}-25 x-75\) is \((x-5)(x-3)(x-2)(x+2.5)\).

Step 1: Identify potential roots

The potential roots of a polynomial can be found as the factors of the constant term. In this case, the constant term is -75. So the factors are \(\pm1, \pm3, \pm5, \pm15, \pm25, \pm75\).

Step 2: Plug in potential roots

Each potential root can be plugged into the original polynomial, if it evaluates to zero, then it is a root. By calculating, we find that -5 and 3 are roots of the polynomial.

Step 3: Polynomial division

Next, we perform synthetic division with the actual polynomial using the roots -5 and 3. When we perform the synthetic division with -5, we get another polynomial \(x^{2}+x-15\), which when divided by 3 results in \(x^{2}+x-5\).

Step 4: Factorize the quotient polynomial

The final polynomial \(x^{2}+x-5\) can be factored further by identifying factors that multiply to -5 and add to 1, which are 2 and -2.5. So it factorizes to become \((x-2)(x+2.5)\).