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Chapter 4: Problem 6

Factor the polynomial completely. $$ 4 k^5-100 k^3 $$

Short Answer

Expert verified
The fully factored form of the given polynomial is \(4 k^3(k - 5)(k + 5)\).

Step by step solution

01

Factor Out the Greatest Common Factor (GCF)

Look for the greatest common factor (GCF) that both terms of the polynomial have. The GCF here is \(4k^3\). Divide each term of the expression by \(4k^3\) to find the remaining part of the polynomial. So the polynomial becomes \(4 k^3(k^2 - 25)\).
02

Factoring the Remaining Polynomial

Next factor the remaining polynomial \(k^2 - 25\). As this is a difference of squares, it can be factored following the rule \(a^2 - b^2 = (a-b)(a+b)\). Accordingly, \(k^2 - 25\) becomes \((k - 5)(k + 5)\).
03

Combine These Factors

Finally, combine these factors to give the fully factored form of the original polynomial. The fully factored form thus is \(4 k^3(k - 5)(k + 5)\).

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Key Concepts

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

Greatest Common Factor
When dealing with polynomials, it's important to start with the basics: identifying the Greatest Common Factor (GCF). The GCF is the largest factor that can divide each term of the polynomial without leaving any remainder. It's like finding the biggest number or term shared among the parts you have.

Let's look at the polynomial from the original exercise: \(4k^5 - 100k^3\). To find its GCF, you follow these simple steps:
  • Compare the coefficients: what is the largest number that divides both 4 and 100? The answer is 4.
  • Look at the variable part: what is the highest power of \(k\) that can divide both \(k^5\) and \(k^3\)? It's \(k^3\).
Therefore, the GCF is \(4k^3\). After factoring out \(4k^3\), you're left with \(4k^3(k^2 - 25)\) in the polynomial. Recognizing and factoring out the GCF simplifies the expression, making the next steps easier.
Difference of Squares
In algebra, recognizing special patterns can save you a lot of time. One of these patterns is the difference of squares.

The difference of squares is a specific type of polynomial that takes the form \(a^2 - b^2\). This can be quickly factored into \((a-b)(a+b)\). It's a neat shortcut because it provides a straightforward way to break down what might look like a complicated expression.
  • When you see something like \(k^2 - 25\), notice that 25 equals \(5^2\).
  • Apply the difference of squares formula: \(k^2 - 5^2 = (k-5)(k+5)\).
This concept is especially handy in exercises like the original one, where after factoring the GCF, we are left with \(k^2 - 25\). Recognizing it as a difference of squares makes the problem much easier to solve.
Algebra 2 Concepts
Algebra 2 introduces many foundational concepts that are pivotal in understanding higher-level math. One major skill is factoring, which involves breaking down complex expressions into simpler ones.

Let's consider how these concepts play out in polynomial factorization from the exercise:
  • Start by identifying and factoring the GCF. This reduces the polynomial to a simpler form.
  • Recognize familiar patterns, like the difference of squares, which allows for quick factoring.
  • Combine the factors to get the fully factored expression.
These factoring techniques are essential tools in algebra. They not only simplify expressions but also give insight into the roots and solutions of equations. Understanding and applying these concepts, such as in the fully factored form \(4k^3(k - 5)(k + 5)\), is crucial for success in Algebra 2 and beyond.

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Most popular questions from this chapter

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