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Chapter 9: Problem 51

Use the Binomial Theorem to expand and simplify the expression. \(3(x+1)^{5}+4(x+1)^{3}\)

Short Answer

Expert verified
The computation results in \(3x^5 + 15x^4 + 34x^3 + 42x^2 + 27x + 7\)

Step by step solution

01

Expand and Simplify the Quintic Term

Firstly, apply the Binomial Theorem to \(3(x+1)^5\). The Binomial Theorem states that: \((x+y)^n\) = \(x^n + C(n, 1)x^{n-1}y + C(n, 2)x^{n-2}y^2 + ... + C(n, n)y^n\), where \(C(n, r)\) represents combinations from n items taken r at a time. So, applying this, we get: \(3(x+1)^5\) = \(3 * (x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1)\)
02

Expand and Simplify the Cubic Term

Next, apply the Binomial Theorem to \(4(x+1)^3\). Proceeding in the same way as step 1, we get: \(4(x+1)^3\) = \(4 * (x^3 + 3x^2 + 3x + 1)\)
03

Distribute the coefficients of each term and combine like terms.

Next, distribute the coefficients 3 and 4 across the terms in the parentheses. Then combine like terms in two previous steps: \(3x^5 + 15x^4 + 30x^3 + 30x^2 + 15x + 3 + 4x^3 + 12x^2 + 12x + 4\). Combining like terms yields: \(3x^5 + 15x^4 + 34x^3 + 42x^2 + 27x + 7\).

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Key Concepts

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

Quintic expansion
The quintic expansion involves applying the Binomial Theorem to expand expressions raised to the fifth power. In our example, we start with the expression \((x+1)^5\).
Using the Binomial Theorem:
  • Identify terms: Here, \(x = x\) and \(y = 1\).
  • Expand each term: The expansion for \((x+1)^5\) is \(x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1\).
  • Apply coefficients: Multiply each term by 3, resulting in \(3x^5 + 15x^4 + 30x^3 + 30x^2 + 15x + 3\).
This method allows us to break down complicated powers into simpler components.
Cubic expansion
Cubic expansion is similar to quintic expansion but involves cubic powers. Here, we examine the expression \((x+1)^3\).
Applying the Binomial Theorem:
  • Identify terms: Here, \(x = x\) and \(y = 1\).
  • Expand each term: The expansion for \((x+1)^3\) is \(x^3 + 3x^2 + 3x + 1\).
  • Apply coefficients: Multiply each term by 4, resulting in \(4x^3 + 12x^2 + 12x + 4\).
This simplifies the process of working with expressions raised to the third power.
Combinations in algebra
Combinations in algebra help us determine the coefficients in the expansion of binomials, and they come from 'n choose k', denoted as \(C(n, k)\) or \(\binom{n}{k}\). This is calculated as:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]Key points to keep in mind:
  • \(C(n, 0)\) and \(C(n, n)\) are always 1.
  • The sum of the exponents in each term of the expansion equals \(n\).
  • Combinations are crucial in finding the coefficients in expansions like quintic and cubic.
Understanding combinations ensures you correctly expand and simplify expressions using the Binomial Theorem.

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Most popular questions from this chapter

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$\frac{1}{2}, \frac{-1}{4}, \frac{1}{8}, \frac{-1}{16}, \dots$$

Use the Binomial Theorem to expand and simplify the expression. \((4 y-3)^{3}\)

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$9,11,13,15,17, \ldots$$

Use the Binomial Theorem to expand and simplify the expression. \((x+1)^{6}\)

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$3,8,13,18,23, \dots$$
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