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Chapter 11: Problem 11

Expand each power. $$ (a-b)^{3} $$

Short Answer

Expert verified
\((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).

Step by step solution

01

Identify the Formula

To expand the expression \((a-b)^3\), we use the binomial theorem, which states that \((a-b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}(-b)^{k}\). Here, \(n=3\), so we need to compute the expansion using this formula.
02

Apply the Binomial Theorem

For \((a-b)^3\), apply the binomial expansion: \[(a-b)^3 = \sum_{k=0}^{3} \binom{3}{k} a^{3-k}(-b)^{k}\]. We need to compute each term in this expansion separately.
03

Calculate Each Term

Compute the terms:- For \(k=0\), the term is \(\binom{3}{0} a^{3}(-b)^{0} = 1 \cdot a^3 = a^3\).- For \(k=1\), the term is \(\binom{3}{1} a^{2}(-b)^{1} = 3a^2(-b) = -3a^2b\).- For \(k=2\), the term is \(\binom{3}{2} a^{1}(-b)^{2} = 3a(-b)^2 = 3ab^2\).- For \(k=3\), the term is \(\binom{3}{3} a^{0}(-b)^{3} = 1(-b)^3 = -b^3\).
04

Combine Terms

Combine all the terms from Step 3 to form the expanded expression:\[(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\].

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Key Concepts

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

Binomial Expansion
Expanding a binomial expression means expressing it as a sum of terms. For example, when you see something like \((a - b)^3\), you can use the Binomial Theorem to break it down into a series of individual components. This is the process of binomial expansion. The Binomial Theorem provides a formula for this expansion:
  • The expression is written as \( (a-b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}(-b)^{k} \).
  • Here, the key elements include \( \binom{n}{k} \), which is a binomial coefficient representing the number of ways to choose \(k\) elements from \(n\) elements.
  • The expression \(a^{n-k}\) represents the decreasing powers of \(a\), and \((-b)^k\) represents the increasing powers of \(-b\).
This theorem is particularly useful for expanding expressions raised to a power, saving you from directly multiplying the binomial multiple times.
Algebraic Expressions
An algebraic expression like \((a-b)^3\) involves numbers, variables, and symbols that represent mathematical operations. These expressions can be quite complex but are a fundamental part of algebra. Understanding how to manipulate them can make tasks such as expansion much simpler. Key aspects include:
  • Variables like \(a\) and \(b\) that can represent unknown numbers.
  • Operations such as addition, subtraction, multiplication, and exponentiation.
  • When expanding expressions, it's crucial to manage these elements systematically.
Breaking down the expression with the binomial theorem helps in managing this complexity by logically structuring how terms contribute to the final polynomial form.
Polynomial Expansion
Polynomial expansion is about rewriting expressions like \((a-b)^3\) as a sum of simpler terms. By applying binomial expansion, we convert a compact expression into a more expanded format. Here鈥檚 how polynomial expansion works:
  • Each term calculated in the expansion has a specific degree, determined by the powers of the variables involved.
  • As seen in \( (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \), the expression is expanded by calculating each possible term.
  • This process reveals how each component contributes to the overall expression by clearly showing the interaction between \(a\) and \(b\).
This expanded form is called a polynomial, and it offers more flexibility for further mathematical operations or simplification. Polynomial expansion is very practical in numerous applications within algebra and calculus.

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Most popular questions from this chapter

Expand each power. $$ \left(\frac{a}{2}+2\right)^{5} $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 4 y-x+y^{2}=1 $$

Prove that each statement is true for all positive integers. $$ 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$

Write a quadratic equation with the given roots. Write the equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are integers. \(6,4\)

OPEN ENDED Write a power of a binomial for which the first term of the expansion is 625\(x^{4}\) .
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