Problem 38 - Precalculus : Building Concepts and Connections

Question

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$x^{4}-6 x^{3}+9 x^{2}-24 x+20 ; \text { zero: } x=5$$

Step-by-Step Solution

Answer

The factorized form of the polynomial after employing the given steps would be: \(x^{4}-6 x^{3}+9 x^{2}-24 x+20 = (x-5)(x-1)^3\).

Step 1: Identify the Known Zero

It's known that \(x=5\) is a zero of the polynomial. The corresponding linear factor is \(x-5\).

Step 2: Apply Polynomial Division

Divide the given polynomial \(x^{4}-6 x^{3}+9 x^{2}-24 x+20\) by the linear factor \(x-5\). Use synthetic division or long division for polynomials, and obtain a simpler polynomial as a result.

Step 3: Solve the Resulting Polynomial

The simpler polynomial obtained in step 2 is a cubic polynomial. To find the remaining linear factors, set this polynomial equal to zero, and find the roots. It may involve techniques such as factoring, use of the quadratic formula, or Rational Root Theorem.

Step 4: Express Original Polynomial as Product of Linear Factors

Combine the linear factor from Step 1 with the linear factors obtained from the zeros in step 3. This will give the original polynomial expressed as a product of linear factors.