Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \(3(x+1)^5\) = \(3 * (x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1)\)

Step 2: 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: \(4(x+1)^3\) = \(4 * (x^3 + 3x^2 + 3x + 1)\)

Step 3: 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: \(3x^5 + 15x^4 + 34x^3 + 42x^2 + 27x + 7\).

Key Concepts

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.