Problem 26 - College Algebra

Question

For the following exercises, expand the binomial. $$(12-4 x)^{2}$$

Step-by-Step Solution

Answer

\((12-4x)^2 = 144 - 96x + 16x^2\).

Step 1: Apply the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 12\), \(b = -4x\), and \(n = 2\).

Step 2: Calculate Each Term Separately

To expand \((12-4x)^2\), calculate each term: - The first term (when \(k = 0\)) is \(\binom{2}{0} (12)^2 (-4x)^0 = 1 \times 144 \times 1 = 144\).- The second term (when \(k = 1\)) is \(\binom{2}{1} (12)^1 (-4x)^1 = 2 \times 12 \times (-4x) = -96x\).- The third term (when \(k = 2\)) is \(\binom{2}{2} (12)^0 (-4x)^2 = 1 \times 1 \times 16x^2 = 16x^2\).

Step 3: Combine the Terms

Add the results from each term to find the expanded expression:1. The first term: 144.2. The second term: \(-96x\).3. The third term: \(16x^2\).Thus, the expansion is \(144 - 96x + 16x^2\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool that allows us to expand expressions raised to a power, such as \((a+b)^n\). It gives a formula to find each term in the expansion.The formula is:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Let's break it down:

• \(n\) is the exponent, telling us how many times the binomial is multiplied by itself.
• \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) items from \(n\) items.
• \(a\) and \(b\) are the terms in our binomial expression, each raised to specific powers based on \(k\).

Using the Binomial Theorem, we can systematically calculate each term to expand any binomial to the required power. In the given exercise, we used this theorem to expand \((12-4x)^2\), leveraging the formula to compute each term separately.Understanding the Binomial Theorem makes it easier to tackle polynomial expansions, especially when the exponent is small, like in this exercise where \(n=2\).

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a polynomial. When we expand a binomial, such as \((12-4x)^2\), we express it as a sum of terms.Each term is carefully calculated by applying the rules of the binomial theorem.This involves multiplying the coefficients, powers of the terms, and paying attention to signs.
Here is how you do it for \((12-4x)^2\):

• Calculate the first term. With \(k=0\), it results in \(12^2\) which equals 144.
• Proceed to the second term. With \(k=1\), calculate using \(12^1(-4x)^1\). This simplifies to \(-96x\).
• The third term considers \(k=2\) and results in \((-4x)^2\), simplifying to \(16x^2\).

Combine these calculated terms to form the polynomial \(144 - 96x + 16x^2\).Polynomial expansion using the binomial theorem is a methodical activity that gives structure to what would otherwise be a lengthy process of manual multiplication and simplification.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations.In the context of our exercise, we started with the binomial expression \((12 - 4x)\) raised to the second power.Expanding this binomial transforms it into a polynomial, another form of an algebraic expression.
Consider the components:

• The terms \(12\) and \(-4x\) in the original binomial are parts of this algebraic expression,where \(12\) is a constant and \(-4x\) is a variable term.
• After expanding, we construct the algebraic expression \(144 - 96x + 16x^2\), which is a polynomial made up of three terms.

This process demonstrates the transition from a simple binomial to a more complex polynomial expression.Understanding algebraic expressions is crucial for solving equations and expressing mathematical relationships clearly.Having a solid grasp of these concepts enables you to manipulate and expand expressions with confidence.