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Chapter 0: Problem 56

Find each product. $$(x-1)^{3}$$

Short Answer

Expert verified
The expanded form of \((x-1)^{3}\) is \(x^3 - 3x^2 + 3x - 1\).

Step by step solution

01

Apply the Binomial Theorem

The binomial theorem can be used to expand the expression \((x-1)^{3}\). The power of 3 indicates that there will be four terms in the expanded form. The expanded form of a binomial \((a-b)^{n}\) is \(a^n - na^{n-1}b + n(n-1)a^{n-2}b^2/2! - n(n-1)(n-2)a^{n-3}b^3/3! + ... \) until \(n\) terms. Here, \(a\) is \(x\), \(b\) is 1, and \(n\) is 3.
02

Expand the Binomial Expression

Substitute \(a\), \(b\), and \(n\) into the binomial theorem and expand the binomial \((x-1)^{3}\). The terms are \(x^3 - 3x^2*1 + 3*2x*1^2/2! - 3*2*1*x^0/3! = x^3 - 3x^2 + 3x - 1\).
03

Simplifying the Expression

Simplify the resulting expression by consolidating like terms, if any. However, in this case, there are no like terms to consolidate, so the simplified form of the expression \((x-1)^{3}\) is \(x^3 - 3x^2 + 3x - 1\).

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\).

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

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?

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