Problem 40 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(4x - 1\right)^3 - 2\left(4x - 1\right)^4 \)

Step-by-Step Solution

Answer

The expanded and simplified form of the given expression using the Binomial Theorem is \( -512x^4 + 576x^3 - 240x^2 + 44x - 3 \).

Step 1: Expanding the first term

Expand \( \left(4x - 1\right)^3 \) using the Binomial theorem. By using the binomial theorem, \( (a + b)^n = \sum{n \choose k} a^{n-k}b^k \), where \(n \choose k\) is binomial coefficient. \n So, \( (4x - 1)^3 = {3 \choose 0} (4x)^3 + {3 \choose 1} (4x)^2 (-1) + {3 \choose 2} (4x)(-1)^2 + {3 \choose 3}(-1)^3 = 64x^3 - 48x^2 + 12x - 1

Step 2: Expanding the second term

Expand by the same procedure, \( -2 \left(4x - 1\right)^4 = -2\left( {4 \choose 0} (4x)^4 + {4 \choose 1} (4x)^3 (-1) + {4 \choose 2} (4x)^2 (-1)^2 + {4 \choose 3}(4x)(-1)^3 + {4 \choose 4} (-1)^4 \right) = -512x^4 + 512x^3 - 192x^2 + 32x - 2

Step 3: Completing the Simplification

Combine the results from step 1 and step 2 to complete the simplified form of the original expression: \( 64x^3 - 48x^2 + 12x - 1 - 512x^4 + 512x^3 - 192x^2 + 32x - 2 \) which simplifies to \( -512x^4 + (64+512)x^3 + (-48-192)x^2 + (12+32)x - 1 - 2\). Hence, the simplified form of the original expression is \( -512x^4 + 576x^3 - 240x^2 + 44x - 3 \).

Key Concepts

Binomial Coefficient

The concept of the binomial coefficient is at the heart of the Binomial Theorem. When you want to expand an expression like \((a + b)^n\), the coefficients of the terms in the expansion are determined using binomial coefficients. These coefficients are represented as \({n \choose k}\), and they are calculated using a factorial formula: \({n \choose k} = \frac{n!}{k!(n-k)!}\).

Think of these coefficients as the multipliers that define how each part of the expansion should "weight" the terms involved. In our example, the expression \((4x - 1)^3\) was expanded using binomial coefficients such as \({3 \choose 0}\), \({3 \choose 1}\), \({3 \choose 2}\), and \({3 \choose 3}\).

Each coefficient is derived based on the number of ways to choose a given number of elements (k) from a set of n elements without regard to order. This provides a way to evenly distribute different combinations of terms in a polynomial expansion.

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a sum of multiple terms. This is achieved through the application of the Binomial Theorem, which states that \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\).

In our example, expanding \((4x - 1)^3\) meant distributing it into simpler terms using the formula above:

• The first term, \(64x^3\), came from \(({3 \choose 0})(4x)^3(-1)^0\).
• Next, \(-48x^2\) resulted from \(({3 \choose 1})(4x)^2(-1)^1\).
• The term \(12x\) came from \(({3 \choose 2})(4x)(-1)^2\).
• The final constant term \(-1\) was derived from \(({3 \choose 3})(-1)^3\).

In essence, each expanded term results from combining exponents and applying the respective binomial coefficient to adjust the contribution of each part of the original binomial.

Simplification

Simplification is the process of combining and reducing the terms in an expression to its simplest form. Once each binomial part is expanded, as we did with \((4x - 1)^3\) and \((4x - 1)^4\), it's time to gather like terms.

The expanded expressions were:

• \((4x - 1)^3 = 64x^3 - 48x^2 + 12x - 1\)
• \(-2(4x - 1)^4 = -512x^4 + 512x^3 - 192x^2 + 32x - 2\)

Combining these results involved careful addition and subtraction of terms with the same power of \(x\).

By adding the coefficients of \(x^3, x^2, x\), and constant terms, the final simplified form of the original expression was reached:

• \(-512x^4\) from the second expansion term alone.
• \((64+512)x^3\) becomes \(576x^3\).
• Adding \(-48x^2\) and \(-192x^2\) gives \(-240x^2\).
• The terms \(12x\) and \(32x\) add up to \(44x\).
• The constant terms \(-1 - 2\) simplify to \(-3\).

Simplification ensures the expression is concise and readable, leaving it in an ideal form for further analysis or evaluation.