In Exercises 15–58, find each product. (x−1)3

Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 56

Chapter 1, Problem 56

In Exercises 15–58, find each product. (x−1)³

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Recognize that the problem involves expanding the cube of a binomial, $(x - 1)^3$. This can be expanded using the Binomial Theorem, which states $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$.

Identify the components of the binomial: $a = x$, $b = -1$, and $n = 3$. Substitute these values into the Binomial Theorem formula.

Write the expanded form of $(x - 1)^3$ using the Binomial Theorem: $\binom{3}{0}x^3(-1)^0 + \binom{3}{1}x^2(-1)^1 + \binom{3}{2}x^1(-1)^2 + \binom{3}{3}x^0(-1)^3$.

Simplify each term by calculating the binomial coefficients $\binom{n}{k}$ and the powers of $-1$. For example, $\binom{3}{0} = 1$, $\binom{3}{1} = 3$, $\binom{3}{2} = 3$, and $\binom{3}{3} = 1$.

Combine the simplified terms to write the expanded polynomial. The result will be in the form $x^3 - 3x^2 + 3x - 1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion is achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This concept is essential for expanding polynomials like (x - 1)^3.

Cubic Functions

A cubic function is a polynomial of degree three, typically expressed in the form f(x) = ax^3 + bx^2 + cx + d. Understanding cubic functions is crucial for recognizing the behavior of the graph, including its turning points and intercepts. In the context of the question, expanding (x - 1)^3 will yield a cubic polynomial.

Factoring and Simplifying Polynomials

Factoring and simplifying polynomials involves rewriting a polynomial as a product of its factors, which can make it easier to analyze or solve. This process often includes identifying common factors or applying special product formulas. In the case of (x - 1)^3, recognizing it as a repeated factor will aid in both expansion and simplification.

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