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Magazine Chapter 13: Problem 26
Use the Binomial Theorem to expand the expression. $$ (1-x)^{5} $$
Short Answer
Expert verified
The expanded expression is \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\).
Step by step solution
01
Identify the Binomial Expansion Formula
The Binomial Theorem states that for any positive integer \( n \), \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our case, we have \(a = 1\), \(b = -x\), and \(n = 5\).
02
Setup the Binomial Coefficients
We need to calculate the binomial coefficients \(\binom{5}{k}\) for \(k = 0\) to \(k = 5\). The formula for binomial coefficients is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
03
Calculate Each Term of the Expansion
For each \(k\) from 0 to 5, calculate \(\binom{5}{k} (1)^{5-k} (-x)^k\) and simplify:- For \(k = 0\): \(\binom{5}{0} (1)^5 (-x)^0 = 1\)- For \(k = 1\): \(\binom{5}{1} (1)^4 (-x)^1 = -5x\)- For \(k = 2\): \(\binom{5}{2} (1)^3 (-x)^2 = 10x^2\)- For \(k = 3\): \(\binom{5}{3} (1)^2 (-x)^3 = -10x^3\)- For \(k = 4\): \(\binom{5}{4} (1)^1 (-x)^4 = 5x^4\)- For \(k = 5\): \(\binom{5}{5} (1)^0 (-x)^5 = -x^5\)
04
Write the Final Expanded Expression
Combine all the terms to form the full expansion: \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
Binomial expansion uses the Binomial Theorem to express a power of a binomial, like \((a + b)^n\), as a sum of terms. Each term involves the original variables raised to successive powers and multiplied by a binomial coefficient.
Binomial expansions simplify complex algebraic expressions by breaking them down into sums.
For the expression \((1-x)^5\), the Binomial Theorem tells us:
Binomial expansions simplify complex algebraic expressions by breaking them down into sums.
For the expression \((1-x)^5\), the Binomial Theorem tells us:
- We start with the binomial formula \((1-x)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-x)^k\).
- This expansion results in multiple terms, specifically six terms (from \(k=0\) to \(k=5\)).
- Each term is simplified individually and then combined into a polynomial expression.
Binomial Coefficients
Binomial coefficients are central to the binomial expansion process. They are the numbers appearing in the expansion of binomial powers and are represented by \(\binom{n}{k}\).
These coefficients are computed using the formula:
These coefficients are computed using the formula:
- \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
- The symbol \(n!\) (n factorial) represents the product of all positive integers up to \(n\).
- For instance, to find \(\binom{5}{3}\), calculate \(\frac{5!}{3!2!} = 10\).
Polynomial Expansion
Polynomial expansion via the Binomial Theorem transforms a simple binomial raised to a power into a polynomial composed of several terms. Each term is a product of a power of the first variable, a binomial coefficient, and a power of the second variable.
For \((1-x)^5\), this leads to expanding it into:
For \((1-x)^5\), this leads to expanding it into:
- \(1\) for \(k=0\)
- \(-5x\) for \(k=1\)
- \(10x^2\) for \(k=2\)
- \(-10x^3\) for \(k=3\)
- \(5x^4\) for \(k=4\)
- \(-x^5\) for \(k=5\)
Algebraic Expressions
Algebraic expressions include variables, constants, and operators like addition and multiplication. The binomial expansion serves as a powerful tool for manipulating these expressions by breaking them down into simpler, manageable parts.
Consider the expression \((1-x)^5\):
Consider the expression \((1-x)^5\):
- It initially looks simple, but its power signifies a more complex structure upon expansion.
- Through binomial expansion, it is transformed into a series of terms (a polynomial), making it easier to analyze or compute substitutions.
- The resulting expression \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\) both encapsulates the original expression's essence and offers a straightforward form for further calculations.
