Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Simplify the expression using the binomial theorem. $$ \frac{(x+h)^{5}-x^{5}}{h} $$

Step-by-Step Solution

Answer

The simplified expression is \(5x^4 + 10x^3 h + 10x^2 h^2 + 5x h^3 + h^4\).

Step 1: Recognize the Binomial Theorem

The binomial theorem states that \((x + h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^k\). Since we need to expand \((x+h)^5\), identify that \(n=5\).

Step 2: Expand Using Binomial Theorem

Expand \((x+h)^5\) using the binomial theorem: \( (x+h)^5 = \binom{5}{0}x^5 h^0 + \binom{5}{1}x^4 h^1 + \binom{5}{2}x^3 h^2 + \binom{5}{3}x^2 h^3 + \binom{5}{4}x^1 h^4 + \binom{5}{5}x^0 h^5 \).

Step 3: Substitute Binomial Expansion

Substitute the expanded form into the given expression: \( \frac{(x+h)^5 - x^5}{h} = \frac{5x^4 h + 10x^3 h^2 + 10x^2 h^3 + 5x h^4 + h^5}{h} \).

Step 4: Simplify the Expression

Factor out \( h \) from the numerator: \( \frac{h(5x^4 + 10x^3 h + 10x^2 h^2 + 5x h^3 + h^4)}{h} \).

Step 5: Cancel Out \( h \)

Cancel \( h \) from the numerator and denominator: \( 5x^4 + 10x^3 h + 10x^2 h^2 + 5x h^3 + h^4 \).

Key Concepts

Polynomial Expansion

In the realm of algebra, polynomial expansion is a vital concept, especially when using the binomial theorem. The binomial theorem provides a formula for expanding expressions of the form \((x + h)^n\), making it easier to handle expressions with higher powers.
To expand \((x+h)^5\), you apply the binomial theorem:

• Each term is formed by choosing \(k\) elements from \(n\), represented as \(\binom{n}{k}\).
• The terms are combined using increasing powers of \(h\) and decreasing powers of \(x\).

So for \((x+h)^5\), it transforms into:

• \(\binom{5}{0}x^5 = x^5\)
• \(\binom{5}{1}x^4h = 5x^4h\)
• \(\binom{5}{2}x^3h^2 = 10x^3h^2\)
• \(\binom{5}{3}x^2h^3 = 10x^2h^3\)
• \(\binom{5}{4}xh^4 = 5xh^4\)
• \(\binom{5}{5}h^5 = h^5\)

These terms help frame the expanded form of a binomial expression, enabling further simplification.

Simplification

Simplification of algebraic expressions involves making them more manageable by reducing their complexity. This is crucial in mathematical operations, especially when working with polynomial expansions.
Once you have the expansion \((x+h)^5 - x^5 = 5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5\), the goal is to simplify:

• Cancel common factors to reduce terms to their simplest form.
• Reduce the expression by grouping and eliminating terms.

For the given problem, after expanding, the expression is \[\frac{5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5}{h}\].
The simplification step is fairly strightforward:

• Factor \(h\) from the numerator, obtaining \(h(5x^4 + 10x^3h + 10x^2h^2 + 5xh^3 + h^4)\).
• Cancel \(h\) from both the numerator and denominator, resulting in the final simplified form \(5x^4 + 10x^3h + 10x^2h^2 + 5xh^3 + h^4\).

This makes the expression straightforward to use in further calculations.

Algebraic Expressions

Algebraic expressions represent combinations of variables and constants, organized through operations like addition, subtraction, multiplication, and division. They are foundational to algebra, allowing us to describe relationships and patterns in a compact form. In this exercise, the expression \(\frac{(x+h)^5-x^5}{h}\) is an excellent example of using algebraic expressions to simplify and solve problems through manipulation.
The manipulation of algebraic expressions often involves:

• Understanding the structure and components of an expression.
• Utilizing algebraic identities like the binomial theorem.
• Applying arithmetic operations to transform and simplify the expressions.

In simplifying the given expression, we combine algebraic techniques with the expansion and reduction processes:

• Firstly, employ the binomial theorem for expansion of \((x+h)^5\).
• Next, carefully perform arithmetic operations to simplify the differences and fractions.
• Finally, utilize cancellation and factorization to reduce the expression, making it more concise and clear for further use.

Mastering the manipulation of algebraic expressions is key to solving a wide range of mathematical problems.