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Chapter 10: Problem 27

Use the binomial theorem to expand the expression. $$(y-2 x)^{4}$$

Short Answer

Expert verified
Expanded form of \((y-2x)^4\) is \(y^4 - 8 y^3 x + 24 y^2 x^2 - 32 y x^3 + 16 x^4\)

Step by step solution

01

Understanding the binomial theorem

The binomial theorem states that for any natural number \(n\), \((a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + b^n\), where \(\binom{n}{r}\) is the binomial coefficient defined as \(n!/(r!(n-r)!)\), and '!' denotes factorial.
02

Identifying the terms in the given binomial

The terms in the binomial are 'y' and '-2x', and the power \(n\) is 4.
03

Expanding the binomial using the binomial theorem

Substitute \(a = y\), \(b = -2x\), and \(n = 4\) into the binomial theorem. This yields:\[(y-2x)^4 = y^4 + \binom{4}{1}y^3(-2x) + \binom{4}{2}y^2(-2x)^2 + \binom{4}{3}y(-2x)^3 + (-2x)^4\]Calculate the binomial coefficients, and simplify the expressions. This results in:\[y^4 - 8 y^3 x + 24 y^2 x^2 - 32 y x^3 + 16 x^4\]

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Key Concepts

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

Binomial Expansion
The binomial expansion allows us to expand expressions of the form \((a + b)^n\). This powerful technique is based on the binomial theorem, which provides a systematic way to express these complex powers as a series of terms. Each term in the expansion has a specific pattern:
  • The first term is simply \(a^n\).
  • Subsequent terms decrease the power of \(a\) and increase the power of \(b\), until \(b^n\).
The expansion of \((a + b)^n\) involves summing up these terms, where each term includes a binomial coefficient. In our example of \((y-2x)^4\), the expansion results in several terms:
  • \(y^4\) as the starting term.
  • \(- 8y^3x\), \(24y^2x^2\), and so forth, until the last term, \(16x^4\).
The process of binomial expansion is fundamental in algebra and is highly useful in simplifying complex algebraic expressions.
Binomial Coefficients
Binomial coefficients are key elements in the binomial theorem, representing the coefficients of the terms in the expansion. These coefficients are expressed as \(\binom{n}{r}\), where \(n\) is the power, and \(r\) indicates the term number.
  • They are calculated using the formula \(\frac{n!}{r!(n-r)!}\).
  • \(n!\) represents "n factorial," which is the product of all positive integers up to \(n\).
In the expansion of \((y-2x)^4\), we calculate binomial coefficients for each term:
  • \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), and so on.
These coefficients multiply the respective terms, influencing their contribution to the overall expansion. Binomial coefficients play a crucial role in determining the structure and pattern of a binomial expansion.
Algebraic Expressions
Algebraic expressions include numbers, variables, and operations like addition and multiplication. When dealing with binomial expansions, they involve combinations of these elements using the rules of algebra.In an expression such as \((y-2x)^4\), each term of the expansion is itself an algebraic expression:
  • Variables \(y\) and \(x\) form the core components.
  • Operations like multiplication and exponentiation apply to these variables.
Understanding algebraic expressions helps in simplifying and manipulating terms obtained through binomial expansion, as encountered in simplified results like \(y^4 - 8y^3x + 24y^2x^2 - 32yx^3 + 16x^4\). This process involves:
  • Applying the binomial coefficients correctly.
  • Careful simplification using algebraic rules.
Mastering algebraic expressions is vital for solving complex mathematical problems and aids in understanding the results of expanded binomials.

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Most popular questions from this chapter

How many different six-letter arrangements are there of the letters in the word PIPPIN? This exercise involves a slightly different strategy than the strategies discussed in the Examples. First draw six slots for six letters. In how many ways can you put the three P's in the slots? You have three slots left over. In how many ways can you place the two I's? The last slot, by default, will contain the N .

State whether the sequence is arithmetic or geometric. $$2,6,18,54, \dots$$

Answer True or False. When picking one coin at random from a bag that contains one quarter, one dime, one nickel, and one penny, "picking a coin with a value of more than one cent" and "picking a penny" are mutually exclusive events.

In Exercises \(5-25,\) prove the statement by induction. $$1+3+3^{2}+\cdots+3^{n-1}=\frac{3^{n}-1}{2}$$

State whether the sequence is arithmetic or geometric. $$0.9,0.81,0.729, \ldots$$
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