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Chapter 11: Problem 18

Use the binomial formula to expand each binomial. $$(x+y)^{8}$$

Short Answer

Expert verified
The expansion is \(x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8\).

Step by step solution

01

Identify the Formula

The binomial expansion formula is given by \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k}\). In this problem, \(a = x\), \(b = y\), and \(n = 8\).
02

Apply the Binomial Coefficients

The coefficients for each term in the expansion are represented by \({8 \choose k}\), where \(k\) runs from 0 to 8. We will use these coefficients in the next steps.
03

Expand the Terms

Consider each term in the expansion as \({8 \choose k} x^{8-k} y^k\) for \(k = 0, 1, 2, \ldots, 8\).
04

Calculate the Coefficients

Compute \({8 \choose k}\) for each \(k\):- \({8 \choose 0} = 1\)- \({8 \choose 1} = 8\)- \({8 \choose 2} = 28\)- \({8 \choose 3} = 56\)- \({8 \choose 4} = 70\)- \({8 \choose 5} = 56\)- \({8 \choose 6} = 28\)- \({8 \choose 7} = 8\)- \({8 \choose 8} = 1\).
05

Form the Expansion

Combine the coefficients with the terms to form the expansion: the expansion is \(x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8\).

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Key Concepts

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

Binomial Theorem
The binomial theorem is a fundamental principle in algebra that helps us expand expressions in the form \((a + b)^n\). In this theorem, each term in the expansion can be described using binomial coefficients, which makes it easier to calculate the powers of a binomial expression. For our given problem, the binomial expansion of \((x + y)^8\) means applying the theorem and using the sum of binomial coefficients to find the expanded terms.
  • The expression consists of two terms: \(x\) and \(y\), raised to the 8th power.
  • We calculate each term's contribution by raising \(x\) to decreasing powers while increasing the power of \(y\), starting from 0 up to 8.
The binomial theorem offers a systematic way to expand expressions, especially useful for large powers.
Binomial Coefficients
Binomial coefficients are central to the binomial theorem. These coefficients determine the proportional weight of each term in the expansion. In our expression \((x + y)^8\), the binomial coefficients are generated by \({n \choose k}\) where \(n\) is the power and \(k\) is the term index.
  • Binomial coefficients can be easily found using Pascal's Triangle or calculations where \({n \choose k} = \frac{n!}{k!(n-k)!}\).
  • They symmetrically increase to a midpoint and then decrease, as seen in the sequence: 1, 8, 28, 56, 70, 56, 28, 8, 1.
These coefficients ensure each term in the expansion is weighted correctly, allowing for accurate representation of \((x + y)^8\) in expanded form.
Combinatorics
Combinatorics plays a crucial role when dealing with binomial expansions. It involves calculating the number of ways to choose elements from a set, which is exactly what binomial coefficients represent. Understanding this concept can make it easier to solve problems involving binomial expansions.
  • In the expression \((x + y)^8\), combinatorics helps us understand the 'choosing' process involved in the indices of terms.
  • The coefficients \({8 \choose k}\) reflect the number of ways to choose \(k\) elements from a set of 8.
This insight is helpful not only in algebra but also in probability, tackling complex combinations and arrangements.
Algebra
Algebra is the language that wraps together all these concepts, giving a structured approach to problem-solving. Using algebraic principles, we can rearrange, expand, and simplify expressions like binomials.
  • The algebraic process of expanding \((x + y)^8\) involves understanding powers, terms, and coefficients.
  • It allows us to express complicated expressions in simpler, expanded forms while obeying the rules of operations and exponents.
Algebra not only involves operations with numbers and symbols but also provides the foundational rules that make solving for the expansion of binomials both possible and systematic.

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