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Chapter 5: Problem 64

Find each product. $$(a-b)\left(a^{2}+b^{2}\right)$$

Short Answer

Expert verified
\[a^{3}-ab^{2}+b^{3}\]

Step by step solution

01

Understand the Problem

Recognize that we need to multiply (a-b) by each term in \((a^{2}+b^{2})\). This is an example of the distributive property.
02

First Multiplication

Multiply the first term in the binomial, which is 'a', with each term in \((a^{2}+b^{2})\). So, it would be \(a*a^{2}+a*b^{2}\).
03

Second Multiplication

Next, multiply the second term in the binomial, which is 'b', by each term in \((a^{2}+b^{2})\). So it would be \(b*a^{2}+b*b^{2}\). Note that we have to respect the signs in each multiplication. Since 'b' is deducted from 'a' (so it's actually -b), we need to keep the minus sign.
04

Simplify

Combine step 2 and step 3, and simplify each term. The result becomes \(a^{3}+ab^{2}-ba^{2}-b^{3}\).
05

Rearrange and Finalize

Rearrange the terms to group the like terms together which would be \(a^{3}-ab^{2}+b^{3}\). This is the final answer.

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Most popular questions from this chapter

Use a vertical format to find each product. $$\begin{aligned}&4 z^{3}-2 z^{2}+5 z-4\\\&3 z-2\end{aligned}$$

Discuss situations in which a vertical format, rather than a horizontal format, is useful for multiplying polynomials.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Other than multiplying monomials, the distributive property is used to multiply other kinds of polynomials.

In each exercise, find the product. $$4 x^{3}\left(4 x^{2}-3 x+1\right)$$

Explain how to convert from scientific to decimal notation and give an example.
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