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Chapter 10: Problem 24

Use the binomial theorem to expand the expression. $$(2 z+y)^{3}$$

Short Answer

Expert verified
The expansion of \( (2z+y)^3 \) using binomial theorem is \(8z^3 + 12z^2y + 12zy^2 + y^3\).

Step by step solution

01

Identify the constituents of the binomial

Identify the constituents of the binomial, in this case, \(a = 2z\) and \(b = y\), and the power \(n = 3\).
02

Apply the binomial theorem

Apply the binomial theorem, which says that the binomial \( (a+b)^n \) can be expanded using the formula \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Using this formula, the binomial \((2z + y)^3\) expands to: \(\binom{3}{0} (2z)^3 y^0 + \binom{3}{1} (2z)^2 y^1 + \binom{3}{2} (2z)^1 y^2 + \binom{3}{3} (2z)^0 y^3\)
03

Calculate each term

Now, calculate each term one by one. After calculating, the expansion becomes: \(8z^3 + 12z^2y + 12zy^2 + y^3\).

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Key Concepts

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

Algebraic Expansion
Algebraic expansion is a process used to simplify expressions raised to a power, such as \((a+b)^n\). This technique essentially "expands" the expression by breaking it down into a sum of terms, each involving powers of the individual components of the binomial. When working with the expression \((2z + y)^3\), algebraic expansion helps us rewrite this compact form into a detailed polynomial:
  • Each term consists of powers of \(2z\) and \(y\).
  • The sum of the exponents in each term equals the original power, here 3.
  • This unpacking provides clarity and a way to work with complex expressions.
This technique is an invaluable tool in algebra, allowing for deeper analysis and easier manipulation of polynomials.
Binomial Coefficient
The binomial coefficient \(\binom{n}{k}\) is central to expanding binomials. These coefficients determine the weight each term carries in the expanded expression. They are represented mathematically as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here鈥檚 how this plays out in \((2z + y)^3\):
  • \(\binom{3}{0} = 1\)
  • \(\binom{3}{1} = 3\)
  • \(\binom{3}{2} = 3\)
  • \(\binom{3}{3} = 1\)
This concept shows up in contexts like counting combinations, and in algebra as coefficients for terms, organizing the expansion systematically.
Polynomial Expressions
Polynomial expressions are sums of terms, where each term is a constant multiplied by variables raised to non-negative integer powers. When expanding a binomial using the binomial theorem, the result is a polynomial:
  • Example: \(8z^3 + 12z^2y + 12zy^2 + y^3\).
  • Each term involves distinct combinations of powers for \(z\) and \(y\).
  • The coefficients, derived from the binomial coefficients and the terms鈥� calculations, are key to each term鈥檚 value.
Understanding polynomial expressions and how they form aids in recognizing patterns and properties, a fundamental skill in algebra and advanced mathematics.

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Most popular questions from this chapter

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