Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Simplify the expression using the binomial theorem. $$ \frac{(x+h)^{4}-x^{4}}{h} $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^3 + 6x^2h + 4xh^2 + h^3\).

Step 1: State the Binomial Theorem

The Binomial Theorem states that \((x + h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}h^k\). We will use this theorem to expand \((x + h)^4\).

Step 2: Expand \((x + h)^4\)

Using the Binomial Theorem: \( (x + h)^4 = \binom{4}{0} x^4h^0 + \binom{4}{1} x^3h^1 + \binom{4}{2} x^2h^2 + \binom{4}{3} x^1h^3 + \binom{4}{4} x^0h^4 \). This simplifies to \( x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).

Step 3: Substitute and Simplify

Substitute the expansion into the expression: \( \frac{(x+h)^4 - x^4}{h} = \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h} \).

Step 4: Cancel Terms

The \(x^4\) terms cancel out, leaving \( \frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h} \).

Step 5: Factor and Simplify Each Term

Factor \(h\) from the numerator: \( \frac{h(4x^3 + 6x^2h + 4xh^2 + h^3)}{h} \). Cancel the \(h\) to get \(4x^3 + 6x^2h + 4xh^2 + h^3\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a technique that allows us to express a power of a binomial, such as \((x + h)^n\), as a sum of terms. Each term is a product of constants and variables raised to appropriate powers. This is where the Binomial Theorem is incredibly helpful.
The Binomial Theorem states that for any positive integer \(n\), the expansion of a binomial \((x + h)^n\) is given by:\[(x + h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}h^k\] This formula tells us how to expand the binomial into multiple terms.

• \(\binom{n}{k}\) is a binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
• \(x^{n-k}h^k\) means that as we progress through the terms, the power of \(x\) decreases while the power of \(h\) increases.

Understanding this theorem allows students to break down complex expressions into simpler, calculable components.

Simplifying Expressions

Simplifying expressions is the process of reducing a complex expression into its simplest form.
Once we have expanded a polynomial using the Binomial Theorem, our next task is to simplify it. This involves combining like terms and performing operations that remove unnecessary elements.
In our example, we expanded \((x + h)^4\), which initially results in:\[ x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \]Before simplifying, we substituted this back into our original expression:\[ \frac{(x+h)^4 - x^4}{h} \] After canceling the \(x^4\) term, the expression simplifies further.

• This means removing the \(x^4\) since it appears in both numerator and subtracted afterward.
• It leads to a simpler expression, \(\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h} \).

Simplifying makes complex algebraic expressions easier to interpret and solve, checking for any chance of reducing it further through cancellation or factoring.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This technique is essential when simplifying higher degree expressions, as it can significantly reduce complexity.
In our example, after expanding and simplifying, we need to factor the expression within the numerator:
\(4x^3h + 6x^2h^2 + 4xh^3 + h^4\).This factorization involves identifying and extracting the greatest common factor, which is \(h\) in this case.
By factoring \(h\) from the expression, we rewrite it as:\[ h(4x^3 + 6x^2h + 4xh^2 + h^3) \]With \(h\) in both the numerator and denominator, we can simplify by canceling \(h\):\[ 4x^3 + 6x^2h + 4xh^2 + h^3 \]Factoring simplifies calculations and is crucial for reducing complex equations, making them easier to solve or analyze.