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Chapter 5: Problem 49

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

Short Answer

Expert verified
The product of the monomial and the polynomial is \(6x^4+8x^3\)

Step by step solution

01

Distribute the monomial to each term

In this step, multiply the monomial \(2x\) by each term in the polynomial \(3x^3+4x^2\). This gives \(2x * 3x^3\) and \(2x * 4x^2\).
02

Simplify the expressions

Next, apply the rule of exponents \(a^n * a^m = a^{n + m}\). This rule applies when you have the same base being multiplied. For \(2x * 3x^3\), multiply the coefficients (2 and 3) and add the exponents (1 from \(x\) and 3 from \(x^3)\) to get \(6x^4\). Similarly, for \(2x * 4x^2\), multiply the coefficients (2 and 4) and add the exponents (1 from \(x\) and 2 from \(x^2)\) to get \(8x^3\).
03

Combine results from the two parts

The last step is to combine the results from step 2 by addition since both terms are being added in the original polynomial. This gives \(6x^4+8x^3\).

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Most popular questions from this chapter

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

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

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I wrote a number where there is no advantage to using scientific notation instead of decimal notation.

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. $$\frac{8 x^{6} y^{3}-12 x^{8} y^{2}-4 x^{14} y^{6}}{-4 x^{6} y^{2}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$(y-1)\left(y^{2}+y+1\right)=y^{3}-1$$
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