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Chapter 5: Problem 71

Find each product. $$\left(x^{2}+1\right)\left(x^{4} y+x^{2}+1\right)$$

Short Answer

Expert verified
\(2x^{6}y + 2x^{4} + 2x^{2} + 1\)

Step by step solution

01

Distribute the First Term of the First Polynomial

Multiply \(x虏\) (the first term of the first polynomial) by each term in the second polynomial (i.e., \(x鈦磞\), \(x虏\), and 1). This gives: \(x^{2} * x^{4}y = x^{6}y\), \(x^{2} * x^{2} = x^{4}\), and \(x^{2} * 1 = x^{2}\).
02

Distribute the Second Term of the First Polynomial

Multiply 1 (the second term of the first polynomial) by each term in the second polynomial (i.e., \(x鈦磞\), \(x虏\), and 1). This gives: \(1 * x^{4}y = x^{4}y\), \(1 * x^{2} = x^{2}\), and \(1 * 1 = 1\).
03

Combine Like Terms

Finally, combine all the resulting terms to find the test product. This results in the expression: \(x^{6}y + x^{4} + x^{2} + x^{4}y + x^{2} + 1\)

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Most popular questions from this chapter

Explain how to multiply polynomials when neither is a monomial. Give an example.

Solve: \(4 x-7>9 x-2 .\) (Section 2.7, Example 7)

Perform the indicated operations. $$(y+6)^{2}-(y-2)^{2}$$

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

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.
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