Problem 44 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$(4 c-3 d)^{4}$$

Step-by-Step Solution

Answer

The expansion of \((4c-3d)^4\) using the binomial theorem is: \((4c-3d)^4 = 256c^4 + 768c^3d + 864c^2d^2 + 432cd^3 + 81d^4\)

Step 1: Identify values for n, a, and b

In our given expression \((4c-3d)^4\), \(n = 4\), \(a = 4c\), and \(b = -3d\).

Step 2: Compute the binomial coefficients

We will compute the binomial coefficients for each \(k\) from \(0\) to \(4\): \({4 \choose 0} = \frac{4!}{0!(4-0)!} = 1\) \({4 \choose 1} = \frac{4!}{1!(4-1)!} = 4\) \({4 \choose 2} = \frac{4!}{2!(4-2)!} = 6\) \({4 \choose 3} = \frac{4!}{3!(4-3)!} = 4\) \({4 \choose 4} = \frac{4!}{4!(4-4)!} = 1\)

Step 3: Apply the binomial theorem to expand the expression

Using our computed binomial coefficients, we can now expand the given expression using the binomial theorem: \[ (4c-3d)^4 = \sum_{k=0}^4 {4 \choose k}(4c)^{4-k}(-3d)^k \] Substituting the binomial coefficients and simplifying each term, we get: \[ (4c-3d)^4 = {4 \choose 0}(4c)^4(-3d)^0 + {4 \choose 1}(4c)^3(-3d)^1 + {4 \choose 2}(4c)^2(-3d)^2 + {4 \choose 3}(4c)^1(-3d)^3 + {4 \choose 4}(4c)^0(-3d)^4 \] \[ (4c-3d)^4 = 1(256c^4)(1) - 4(64c^3)(-3d) + 6(16c^2)(9d^2) - 4(4c)(-27d^3) + 1(1)(81d^4) \]

Step 4: Simplify the expression

Now, we just need to simplify the expanded expression: \[ (4c-3d)^4 = 256c^4 - 4(-192c^3d) + 6(144c^2d^2) - 4(-108cd^3) + 81d^4 \] \[ (4c-3d)^4 = 256c^4 + 768c^3d + 864c^2d^2 + 432cd^3 + 81d^4 \] So the expansion of \((4c-3d)^4\) using the binomial theorem is: \[ (4c-3d)^4 = 256c^4 + 768c^3d + 864c^2d^2 + 432cd^3 + 81d^4 \]

Key Concepts

Binomial Expansion

The binomial expansion is a method to expand expressions raised to a power. When you see something like \((a + b)^n\), that's where the binomial expansion comes into play. This process allows us to express the power as a sum of terms, using a series of calculations based on a well-known formula known as the binomial theorem.

The expansion involves the use of binomial coefficients and the two terms \(a\) and \(b\) raised to suitable powers. By consistently applying the theorem, which states that:

• \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\)

This allows you to rearrange an original expression into an expanded form.

For example, instead of calculating \((4c - 3d)^4\) directly, we can break it into simpler pieces using the expansion, ultimately making calculations manageable and less error-prone.

Binomial Coefficients

Binomial coefficients are essential parts of the binomial expansion. They determine how combinations of terms will appear in the expanded expression. Imagine them as specific weights that tell how many times each term will show up as you expand.

• Usually denoted as \({n \choose k}\), where \(n\) is the power to which the binomial is raised, and \(k\) is the term position in the expansion, starting from zero.
• The formula to calculate a binomial coefficient is: \(\frac{n!}{k!(n-k)!}\).

By employing this coefficient, you can understand how to build the terms in the expansion. For example, in expanding \((4c - 3d)^4\), each term utilizes a binomial coefficient that multiplies the respective powers of \(4c\) and \(-3d\). These coefficients ensure that the resulting polynomial is correctly balanced in size and power.

Polynomial Expansion

Polynomial expansion is the transformation of an expression like \((a + b)^n\) into a sum of terms without parentheses, typically in a polynomial form. This process broadens your understanding of distributing expressions, allowing each part of the original binomial to contribute to all parts of the expanded polynomial.

• When expanded, you get terms like a combination of \(a\) and \(b\), each raised to different powers but combined in a single expression.
• These expanded terms often have coefficients that were determined during the binomial coefficient calculation process.

Understanding polynomial expansion will help you grasp how complex expressions result in simpler polynomial forms, like turning \((4c-3d)^4\) into the detailed polynomial \(256c^4 + 768c^3d + 864c^2d^2 + 432cd^3 + 81d^4\). This result showcases the orderly manner in which each combination of the binomial's parts contributes to the final polynomial.