Problem 72 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The completely factored form of the polynomial is \((x+5)(x-5)(x+3)\).

Step 1: Apply the Rational Root theorem

The Rational Root Theorem postulates that if a polynomial function has integer coefficients, then every rational zero will have the form \(\frac{p}{q}\). 'p' is a factor of the constant term and 'q' is a factor of the leading coefficient. For the given polynomial, \(x^{3}+3x^{2}-25x-75\), the constant term is -75 and the leading coefficient is 1. So possible rational roots are \(\pm\) factors of 75 . After testing all possible roots, one of the roots is found to be -5.

Step 2: Polynomial division

Once the root is found, perform polynomial division or synthetic division of the given polynomial by \((x+5)\). The quotient you get is \(x^2-2x-15\).

Step 3: Factor the quadratic polynomial

Factor the quadratic quotient using standard techniques. The quadratic polynomial \(x^2-2x-15\) factors to \((x-5)(x+3)\).

Step 4: Write the completely factored form

The original polynomial factors as \((x+5)(x-5)(x+3)\).