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Chapter 5: Problem 47

Find each product of the monomial and the polynomial. $$2 y^{2}\left(3 y^{2}-4 y+7\right)$$

Short Answer

Expert verified
\(6y^{4} - 8y^{3} + 14y^{2}\)

Step by step solution

01

Identify the Terms

The monomial is \(2y^{2}\) and the polynomial is \(3y^{2} - 4y + 7\). Each term within the brackets will be multiplied by the monomial. This implies the operations will be: \(2y^{2} \times 3y^{2}\), \(2y^{2} \times -4y\), and \(2y^{2} \times 7\).
02

Multiply Monomial with each Term in Polynomial

Multiply the monomial with each term inside the polynomial brackets as follows: \[\begin{align*}2y^{2} \times 3y^{2} &= 6y^{4} \2y^{2} \times -4y &= -8y^{3} \ 2y^{2} \times 7 &= 14y^{2}\end{align*}\]
03

Combine the Results

Combine the results from Step 2 to get the final answer: \[6y^{4} - 8y^{3} + 14y^{2}\]

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