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Chapter 1: Problem 177

Exercises \(177-179\) will help you prepare for the material covered in the next section. Factor completely: \(x^{3}+x^{2}-4 x-4\)

Short Answer

Expert verified
The polynomial \(x^{3}+x^{2}-4 x-4\) can be completely factored as \((x + 1)(x + 2)(x - 2)\)

Step by step solution

01

Group Terms

Group terms that appears to have common factors. Do this by looking at the pairs of terms that might add up to give the polynomial. In this case, it's natural to group the first two terms and the last two: \((x^{3} + x^{2}) - (4x + 4)\)
02

Factor out common factors

Factor out any common factors in each of the groups. In the first group \(x^{3} + x^{2}\), \(x^{2}\) can be factored out, and in the second group, a factor of 4 can be pulled out. The grouped polynomial becomes \(x^{2}(x + 1) - 4(x + 1)\)
03

Factor the polynomial

Now, as you see, we have two terms and both terms have a common factor \(x + 1\). This can be factored out of the entire expression. So, complete the factoring by pulling out this common factor. The factored form of the polynomial is \((x + 1)(x^{2} - 4)\)
04

Check if it can be factored more

See if any of the remaining terms can be factored more. In this case, the term \(x^{2}-4\) is a difference of squares, which can be factored into \((x+2)(x-2)\). So, the fully factored form of the polynomial is \((x + 1)(x + 2)(x - 2)\)

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