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Chapter 10: Problem 34

Factor the expression completely. \(t^{3}-t^{2}-16 t+16\)

Short Answer

Expert verified
The completely factored form of the given polynomial \(t^{3}-t^{2}-16 t+16\) is \((t-1)(t-4)(t+4)\)

Step by step solution

01

Divide the given polynomial into two groups

The polynomial we have is \(t^{3}-t^{2}-16 t+16\). We divide this into two groups as \(t^{3}-t^{2}\) and \(-16t+16\).
02

Factor out the greatest common factor from each group

In the first group \(t^{3}-t^{2}\), \(t^{2}\) is the common factor. Factoring out, we get \(t^{2}(t-1)\). In the second group \(-16t + 16\), -16 is the common factor. So, we get \(-16(t-1)\). The polynomial now reads as \(t^{2}(t - 1) - 16(t - 1)\). Note that \(t-1\) is a common factor.
03

Factor out the common binomial

Since \(t-1\) is now a common factor in both terms, it's factored out. This gives us \((t-1)(t^{2} - 16)\)
04

Continue factoring until the polynomial is completely factored

Looking at the second factor \(t^{2} - 16\), it is recognized as a difference of squares which can be factored as \((t-4)(t+4)\). So finally, the completely factored form of given expression is \((t-1)(t-4)(t+4)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
When you see an expression such as \(t^2 - 16\), you might notice it fits a special pattern called the difference of squares. A difference of squares is when you have something of the form \(a^2 - b^2\). This can be factored using the identity: \(a^2 - b^2 = (a-b)(a+b)\).

In simple terms, it is two perfect squares separated by a minus sign. Recognizing this pattern is crucial because it helps in simplifying expressions quickly.
  • For \(t^2 - 16\), \(a\) is \(t\) and \(b\) is \(4\) (since \(4^2 = 16\)).
  • Applying the formula gives us \((t - 4)(t + 4)\).
Hence, this technique drastically reduces complexity, turning complicated expressions into simpler, easier-to-manage factors.
Greatest Common Factor (GCF)
Finding the greatest common factor (GCF) is all about identifying the largest factor that divides each term in an expression. It's like finding your common denominator, but for polynomial terms.

In our example, let's look at \(t^3 - t^2\):
  • Each term has a common factor of \(t^2\).
  • Factoring this out gives \(t^2(t-1)\).
Similarly, for the second group \(-16t + 16\):
  • Both terms share a common factor of \(-16\).
  • This simplifies to \(-16(t-1)\).
Extracting the GCF helps break down expressions into simpler parts that further reveal hidden structures like the common binomial in this example.
Factoring by Grouping
Factoring by grouping is a strategy used when dealing with four-term polynomials, like in our problem \(t^3 - t^2 - 16t + 16\). This method involves dividing terms into manageable pairs to identify and extract common elements.

Begin by grouping terms: divide the polynomial into \(t^3 - t^2\) and \(-16t + 16\). Once grouped, you find the GCF of each pair:
  • From \(t^3 - t^2\), take out \(t^2\), resulting in \(t^2(t-1)\).
  • From \(-16t + 16\), extract \(-16\), which gives you \(-16(t-1)\).
Notice how both results contain \(t-1\) as a factor. This common binomial can then be factored out, leading to one step closer to the full solution: \((t-1)(t^2 - 16)\).

Factoring by grouping is powerful as it reveals common factors in complex polynomials, easing further simplification steps.

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Most popular questions from this chapter

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