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Chapter 5: Problem 46

Find each product of the monomial and the polynomial. $$4 y^{2}\left(y^{2}+2 y\right)$$

Short Answer

Expert verified
The product of the monomial \(4y^{2}\) and the polynomial \(y^{2}+2y\) is \(4y^{4}+8y^{3}\).

Step by step solution

01

Identify the monomial and the polynomial

Start by identifying the monomial and the polynomial. The monomial here is \(4y^{2}\) and the polynomial is \(y^{2}+2y\). We want to distribute the monomial across every term in the polynomial.
02

Multiply the monomial with each term of the polynomial

Simply multiply the monomial \(4y^{2}\) first with \(y^{2}\) and then \(2y\). This will give us two new terms: \(4y^{2} \cdot y^{2}\) and \(4y^{2} \cdot 2y\). When doing the multiplication, remember the rule for multiplying terms with the same bases: you add the exponents.
03

Compute the multiplications

Perform the multiplications: \(4y^{2} \cdot y^{2} = 4y^{4}\) and \(4y^{2} \cdot 2y = 8y^{3}\). At this step, subtract the indices while multiplying the coefficients.
04

Write down the final solution

Finally, write down the sum of the two new terms to get: \(4y^{4}+8y^{3}\). This is the result of multiplying the given monomial and polynomial.

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Most popular questions from this chapter

Explain the product rule for exponents. Use \(2^{3} \cdot 2^{5}\) in your explanation.

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

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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5^{-2}>2^{-5}$$

Simplify: \(24+8 \cdot 3+28 \div(-7)\).
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