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Chapter 5: Problem 72

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

Short Answer

Expert verified
The product of the given polynomials is \(x^{3}y^{4} + x^{2}y^{2} + x^{2}+ x y^{4}+ y^{2}+ 1\).

Step by step solution

01

Distribute the first terms

Multiply the first term of the first bracket \(x^{2}\) with every term in the second bracket \((x y^{4}+y^{2}+1)\) which yields \(x^{3}y^{4} + x^{2}y^{2} + x^{2}\)
02

Distribute the second terms

Multiply the second term of the first bracket \(1\) with every term in the second bracket \((x y^{4}+y^{2}+1)\) which gives \(x y^{4} + y^{2} + 1\)
03

Combine the terms

Combine the terms from Step 1 and Step 2. This gives the final expression \(x^{3}y^{4} + x^{2}y^{2} + x^{2}+ x y^{4}+ y^{2}+ 1\)

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Most popular questions from this chapter

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

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.

Use a vertical format to find each product. $$\begin{array}{l}x^{2}+6 x-4 \\\x^{2}-x-2 \\\\\hline\end{array}$$

In Exercises \(53-78,\) divide the polynomial by the monomial. Check each answer by showing that the product of the divisor and the quotient is the dividend. $$\frac{20 x^{7} y^{4}-15 x^{3} y^{2}-10 x^{2} y}{-5 x^{2} y}$$

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