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Chapter 5: Problem 54

Find each product of the monomial and the polynomial. $$-6 x^{2}\left(3 x^{2}-2 x-7\right)$$

Short Answer

Expert verified
\(-18x^4 + 12x^3 + 42x^2\

Step by step solution

01

Identify the Monomial and Polynomial

The monomial is \(-6x^2\) and the polynomial is \(3x^2 - 2x - 7\). We're going to distribute, or multiply, the monomial across every term in the polynomial.
02

Apply the Distributive Property - First Term

Multiply the monomial \(-6x^2\) by the first term of the polynomial \(3x^2\). Remember that when multiplying terms with the same base, you add the exponents. So, \(-6x^2 * 3x^2\) becomes \(-18x^4\).
03

Apply the Distributive Property - Second Term

Then, multiply the monomial \(-6x^2\) by the second term in the polynomial \(-2x\). So, \(-6x^2 * -2x\) becomes \(12x^3\)
04

Apply the Distributive Property - Third Term

Lastly, multiply the monomial \(-6x^2\) by the third term in the polynomial \(-7\). This gives \(-6x^2 * -7\) or \(42x^2\).
05

Write Out the Expanded Form

Now combine all the results from the distributive steps. The expanded form of the initial product is \(-18x^4 + 12x^3 + 42x^2\).

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Most popular questions from this chapter

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

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

What polynomial, when divided by \(3 x^{2}\), yields the trinomial \(6 x^{6}-9 x^{4}+12 x^{2}\) as a quotient?

Explain the negative exponent rule and give an example.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm working with two monomials that I can add, although I cannot multiply them.
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