Problem 94 - Introductory Algebra for College Students

Question

Factor each polynomial. $$7 x^{5}-7 x^{4}+x^{3}-x^{2}+3 x-3$$

Step-by-Step Solution

Answer

The factoring of the polynomial \(7x^5 - 7x^4 + x^3 - x^2 + 3x - 3\) is \((x - 1)(7x^4 + x^2 + 3)\).

Step 1: Identifying Common Factors

First, we can see that each term in the polynomial has at least one factor of \(x\) and they all can be divided by 1, so we don't have any common factor to factor out initially.

Step 2: Grouping and Factoring

Since there isn't any common factor for all the terms of the polynomial, let's try to group the similar terms together. Starting from the first term, group it with the next term that contains the same power of \(x\). Do the same for the rest of the terms. This gives us: \(\{7x^5 - 7x^4\} + \{x^3 - x^2\} + \{3x - 3\}\). Now, factor out the common factors in each pair. This results in: \(7x^4(x - 1) + x^2(x - 1) + 3(x - 1)\).

Step 3: Final Factoring

As seen from the previous step, \((x - 1)\) is a mutual term in all three. Thus, we can factor out \((x - 1)\) and group the other terms together, which gives: \((x - 1)(7x^4 + x^2 + 3)\).