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Chapter 0: Problem 54

Find each product. $$(3 x+4)^{3}$$

Short Answer

Expert verified
The product of the given expression \((3x+4)^3\) is \(27x^3 + 108x^2 + 144x + 64\).

Step by step solution

01

Understand the Binomial theorem

The Binomial theorem allows us to expand expressions of the form \((a+b)^n\). For the cubic case \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). The coefficients 1, 3, 3, 1 are from the third row of Pascal's triangle.
02

Identify the terms in the expression

For the given expression \((3x+4)^3\), \(a = 3x\) and \(b = 4\).
03

Expand using the Binomial theorem

Substitute \(a = 3x\) and \(b = 4\) into the equation for the cubic binomial expansion. This yields \((3x)^3 + 3*(3x)^2*4 + 3*3x*(4)^2 + 4^3\).
04

Simplify the expression

Simplify each term to find the expanded form of \((3x+4)^3\). Which equals \(27x^3 + 108x^2 + 144x + 64\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{2}+36=(x+6)^{2}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?
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