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Chapter 5: Problem 57

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

Short Answer

Expert verified
The product is \(x^{2} - y^{4}\)

Step by step solution

01

(First)

First, we will multiply the first terms in each binomial. So, we will have \(x * x\) which makes \(x^{2}\).
02

(Outer)

Second, we will multiply the outermost terms, that is \(x * -y^{2}\). Which gives us \(-x*y^{2}\).
03

(Inner)

The next step is to multiply the innermost terms, i.e., \(y^{2} * x\). This results in \(x*y^{2}\).
04

(Last)

Finally, we multiply the last terms of each binomial, \(y^{2} * -y^{2}\), resulting in \(-y^{4}\).
05

(Combine)

After applying FOIL, we now combine the results: \(x^{2} - x*y^{2} + x*y^{2} - y^{4}\). Here, we can see that the \(x*y^{2}\) terms cancel out as they are opposites, yielding a simplified equation.

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Most popular questions from this chapter

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

Find each of the products in parts (a)-(c). a. \((x-1)(x+1)\) b. \((x-1)\left(x^{2}+x+1\right)\) c. \((x-1)\left(x^{3}+x^{2}+x+1\right)\) d. Using the pattern found in parts (a)-(c), find $(x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right) without actually multiplying.

Explain the power rule for exponents. Use \(\left(3^{2}\right)^{4}\) in your explanation.

We have seen that in \(2009,\) the United States government spent more than it had collected in taxes, resulting in a budget deficit of \(\$ 1.35\) trillion. If there are approximately \(3.2 \times 10^{7}\) seconds in a year, approximately how many years is 1.35 trillion seconds? (Note: 1.35 trillion seconds would take us back in time to a period when Neanderthals were using stones to make tools.)

Explain how to simplify an expression that involves a quotient raised to a power. Provide an example with your explanation.
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