Problem 26 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(c + d\right)^3 \)

Step-by-Step Solution

Answer

The expanded and simplified expression of \( \left(c + d\right)^3 \) is \(c^3 + 3c^2d + 3cd^2 + d^3\).

Step 1: Recall the Binomial Theorem

The Binomial Theorem states that \( \left(c + d\right)^n \), where n is a positive integer, is the sum of \( \binom{n}{k} (c^{n-k}) (d^k) \) for k = 0 to n. Where \( \binom{n}{k} \) represent the binomial coefficients, often represented graphically in Pascal's Triangle.

Step 2: Apply the Binomial Theorem to the problem

In the given expression, \( \left(c + d\right)^3 \), c and d are the two terms of the binomial and the exponent is 3. We apply the Binomial Theorem to expand it as follows: (1) For k=0: \( \binom{3}{0} (c^{3-0}) (d^0) = 1*c^3*d^0 =c^3 \) (2) For k=1: \( \binom{3}{1} (c^{3-1}) (d^1) = 3*c^2*d \) (3) For k=2: \( \binom{3}{2} (c^{3-2}) (d^2) = 3*c*d^2 \) (4) For k=3: \( \binom{3}{3} (c^{3-3}) (d^3) = 1*c^0*d^3 =d^3 \)

Step 3: Combine the terms

The expanded expression becomes the sum of the terms obtained for each k. Therefore, \( \left(c + d\right)^3 = c^3 + 3c^2d + 3cd^2 + d^3 \).

Key Concepts

Binomial Coefficients

Understanding binomial coefficients is crucial when it comes to expanding expressions using the Binomial Theorem. These coefficients are the numerical factors that multiply the terms in the expansion of binomials like \( (c + d)^n \) where \( n \) is a non-negative integer.

Binomial coefficients for a given power of \( n \) are determined by the formula \( \binom{n}{k} \) which is read as \

Pascal's Triangle

Pascal's Triangle offers a simple yet powerful way to visualize and determine binomial coefficients without resorting to calculations. It is a triangular array of numbers where each number is the sum of the two numbers directly above it from the previous row. The edge numbers are always 1, and by construction, the triangle showcases the coefficients of the binomial expansion.

To use Pascal's Triangle for our exercise \( (c + d)^3 \) we simply look at the fourth row, considering the first row as row zero. The numbers in this row are 1, 3, 3, and 1, which correspond to the coefficients in the expansion:

• For \( k=0 \) the coefficient is 1, giving us the term \( c^3 \)
• For \( k=1 \) the coefficient is 3, giving us the term \( 3c^2d \)
• For \( k=2 \) the coefficient is 3, resulting in \( 3cd^2 \)
• For \( k=3 \) the coefficient is 1, resulting in \( d^3 \)

Polynomial Expansion

Let's delve into polynomial expansion using the Binomial Theorem. Expanding a polynomial means expressing it as the sum of its terms, raised to various powers, multiplied by the respective binomial coefficients. The Binomial Theorem provides a structured approach for these expansions.

In our exercise where we expand \( (c + d)^3 \) we use the theorem which tells us that each term in the expansion is of the form \( \binom{n}{k} c^{n-k} d^k \) for \( k \) from 0 to \( n \). Breaking down the polynomial \( (c + d)^3 \) into individual terms, we apply the theorem and, combining the terms, obtain the fully expanded polynomial:\[ c^3 + 3c^2d + 3cd^2 + d^3 \].

By understanding this process, students can tackle a wide range of problems involving polynomial expansion and develop a stronger grasp of algebraic expressions.