Problem 48 - Algebra and Trigonometry

Question

Simplify 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: Apply the Binomial Theorem

We start by expanding \((x+h)^4\) using the Binomial Theorem. The Binomial Theorem states that for any positive integer \(n\), \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] In this case, \((x+h)^4\) can be expanded as:\[(x+h)^4 = \binom{4}{0} x^4 h^0 + \binom{4}{1} x^3 h^1 + \binom{4}{2} x^2 h^2 + \binom{4}{3} x^1 h^3 + \binom{4}{4} h^4\] Simplifying the terms, we get:\[x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\]

Step 2: Substitute and Simplify the Expression

Substitute the expansion from Step 1 into the original expression:\[\frac{(x+h)^4 - x^4}{h} = \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h}\] Cancel out \(x^4\) from the numerator:\[\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}\]Factor \(h\) out of the numerator:\[\frac{h(4x^3 + 6x^2h + 4xh^2 + h^3)}{h}\]

Step 3: Simplify the Fraction

Cancel out \(h\) from the numerator and the denominator:\[4x^3 + 6x^2h + 4xh^2 + h^3\] This is the simplified form of the original expression.

Key Concepts

Polynomial Expansion

Polynomial expansion allows us to express expressions like \((a+b)^n\) in terms of simpler components. This is useful in algebra to simplify expressions or solve equations. The binomial theorem is a primary tool for expanding such polynomial expressions.

The binomial theorem provides a formula:

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

where

• \(\binom{n}{k}\) is the binomial coefficient, which tells us how many ways we can choose \(k\) elements from a set of \(n\) elements.

For example, when expanding \((x+h)^4\), you use the theorem to express it as \(x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\).

This allows you to view the polynomial as a sum of constant times powers of \(x\) and \(h\), making further manipulation and simplification easier.

Simplification

Simplification involves reducing expressions to their simplest form, making them easier to handle or solve. Once expanded, polynomials can often be simplified by combining like terms or factoring.

In this particular exercise, after we expand \((x+h)^4\), we subtract \(x^4\) from it. This step cancels out the \(x^4\) term, simplifying the expression significantly:

• \(\frac{(x+h)^4 - x^4}{h}\) becomes \(\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}\)

To simplify further, factor an \(h\) from the numerator, which allows you to cancel an \(h\) from the numerator and the denominator. This makes the expression more succinct and manageable. The result is \(4x^3 + 6x^2h + 4xh^2 + h^3\), which is both simpler and easier to interpret.

Algebraic Fractions

Algebraic fractions consist of polynomials in the numerator and/or the denominator. Simplifying them often requires factoring or canceling terms.

When simplifying algebraic fractions, it's key to look for common factors across the numerator and denominator. In this exercise, the expression originally \(\frac{(x+h)^4 - x^4}{h}\) is a perfect place to apply this concept.

Once expanded and terms like \(x^4\) are canceled out, we ended up with:

• \(\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}\)

This prompts a factoring of the numerator, which has a common factor of \(h\), allowing it to be factored out:

• \(h(4x^3 + 6x^2h + 4xh^2 + h^3)\)

By canceling the \(h\) that's common in the numerator and denominator, the fraction simplifies to:

• \(4x^3 + 6x^2h + 4xh^2 + h^3\)

This process is vital in making complex algebraic fractions more understandable and easier to work with.