Problem 93 - Introductory Algebra for College Students

Question

Factor each polynomial. $$3 x^{5}-3 x^{4}+x^{3}-x^{2}+5 x-5$$

Step-by-Step Solution

Answer

\((x - 1)(3x^4 + x^2 + 5)\)

Step 1: Simplification

The first step is to simplify each term in the polynomial by writing the coefficients and variables separately. Here, the polynomial can be rewritten as \(3x^5 - 3x^4 + x^3 - x^2 + 5x - 5\) as \((3x^5 - 3x^4) + (x^3 - x^2) + (5x - 5)\).

Step 2: Factorize

The next step is to factor out the common factors from each pair of terms. That gives: \(3x^4(x - 1) + x^2(x - 1) + 5(x - 1)\). Now, you can observe that \((x - 1)\) is a common factor.

Step 3: Rearranging

Now, we rearrange and factor out \((x - 1)\). Here, the final factorised form of the polynomial is:\((x - 1)(3x^4 + x^2 + 5)\)