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Chapter 13: Problem 17

Use the binomial theorem to expand each binomial. $$(m-n)^{3}$$

Short Answer

Expert verified
The expansion of \((m - n)^3\) is \(m^3 - 3m^2n + 3mn^2 - n^3\).

Step by step solution

01

State the Binomial Theorem

The Binomial Theorem is given by \[(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \text{...} + \binom{n}{n}a^0b^n\] where \(\binom{n}{k}\) is a binomial coefficient.
02

Identify \(a\), \(b\), and \(n\)

In the expression \((m - n)^3\), identify \(a = m\), \(b = -n\), and \(n = 3\).
03

Write down the general expansion

Using the Binomial Theorem for \((m - n)^3\), we have:\[(m - n)^3 = \binom{3}{0}m^3(-n)^0 + \binom{3}{1}m^2(-n)^1 + \binom{3}{2}m^1(-n)^2 + \binom{3}{3}m^0(-n)^3\]
04

Simplify the binomial coefficients

Calculate the binomial coefficients:\(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\)
05

Substitute and expand

Substitute the binomial coefficients and simplify the powers of \(-n\):\[(m - n)^3 = 1 \times m^3 \times (-n)^0 + 3 \times m^2 \times (-n)^1 + 3 \times m \times (-n)^2 + 1 \times (-n)^3\]This simplifies to:\[m^3 - 3m^2n + 3mn^2 - n^3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

binomial coefficient
The binomial coefficient, denoted as \(\binom{n}{k}\), is a key part of the Binomial Theorem. It represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order. The binomial coefficient can be calculated using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] where \(n!\) (n factorial) is the product of all positive integers up to \(n\). For example, \(\binom{3}{1}\) tells us how many ways we can choose one element out of three, which is 3.
polynomial expansion
Polynomial expansion refers to expressing a binomial raised to a power as a sum of terms. Using the Binomial Theorem, we can expand expressions like \((m-n)^3\) into simpler terms. The theorem states that \((a + b)^n\) can be expanded into a series where each term is composed of a binomial coefficient, powers of \(a\), and powers of \(b\). For example, \((m-n)^3\) expands to \((m-n)^3 = \binom{3}{0}m^3(-n)^0 + \binom{3}{1}m^2(-n)^1 + \binom{3}{2}m^1(-n)^2 + \binom{3}{3}m^0(-n)^3\).
algebraic expressions
Algebraic expressions consist of variables, numbers, and operations (such as addition and multiplication). They can be simple, like \(m + n\), or more complex, like \((m - n)^3\). When simplifying algebraic expressions, it鈥檚 important to follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction). Understanding how to manipulate and expand these expressions is a fundamental skill in algebra.
exponents
Exponents indicate how many times a number (the base) is multiplied by itself. For example, \(m^3\) means \(m \times m \times m\). When expanding binomials, you work with exponents to express the repeated multiplication of each term. In \((m-n)^3\), the exponents descend for \(m\) and ascend for \((-n)\). This helps to organize the polynomial expansion into a series of terms with different powers.
simplifying expressions
Simplifying expressions means to rewrite them in a more compact and understandable form. This involves combining like terms (terms with the same variables raised to the same power) and performing arithmetic operations. For the expansion \((m-n)^3 = m^3 - 3m^2n + 3mn^2 - n^3\), each term is simplified by multiplying the coefficients and applying the exponents. The final simplified form is easier to interpret and use in further calculations.

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Most popular questions from this chapter

Evaluate each expression. $$\frac{9 !}{2 ! 7 !}$$

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