Question

Which of the following is the complete factorization of \(x^{3}-5 x^{2}+4 x-20 ?\) A. \((x+2)(x+2)(x-5)\) B. \((x+2)(x-2)(x-5)\) C. \(\left(x^{2}+4\right)(x-5)\) D. \((x-4)(x-1)(x-20)\)

Short answer

The correct answer is Option B: \((x+2)(x-2)(x-5)\). It completely factorizes the given polynomial.

Step-by-step solution

Set the equation to zero

Firstly, set the polynomial equation to zero: \(x^{3}-5x^{2}+4x-20=0\). This will help us to find the factors of the polynomial.

Testing Options using synthetic division

Use synthetic division to find a root. Choose a potential root (For example, using Option A, we may choose -2). Set up synthetic division with coefficients of original polynomial and perform division.

Evaluate the remainder

If the remainder is zero, the selected number is a factor of the polynomial, otherwise, choose another potential factor/root and repeat Step 2. Continue this process until all factors are found.

Identify answer from options

After identifying the factors, check each option to see which one matches with your derived factors. Remember that the factors can be in different orders, but they should be the same when expanded. If none of the options matches with your derived factors, the provided options might have missed the correct factorization.