Problem 56 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{(x+h)^{3}+5(x+h)-\left(x^{3}+5 x\right)}{h}$$

Step-by-Step Solution

Answer

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

Step 1: Expand \((x+h)^3\)

We start by expanding the expression \((x+h)^3\). Use the binomial theorem:\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\].

Step 2: Distribute 5 in \(5(x+h)\)

Distribute 5 to each term in \(5(x+h)\):\[5(x+h) = 5x + 5h\].

Step 3: Simplify the numerator

Substitute the expanded expressions from Step 1 and Step 2 into the numerator:\[(x^3 + 3x^2h + 3xh^2 + h^3) + (5x + 5h) - (x^3 + 5x)\].Distribute the subtraction:\[x^3 + 3x^2h + 3xh^2 + h^3 + 5x + 5h - x^3 - 5x\].Simplify by canceling terms:\[3x^2h + 3xh^2 + h^3 + 5h\].

Step 4: Factor common terms from the numerator

Factor \(h\) out of each term in the simplified expression :\[h(3x^2 + 3xh + h^2 + 5)\].

Step 5: Simplify the expression

Since \(heq 0\), cancel \(h\) in the numerator and denominator:\[\frac{h(3x^2 + 3xh + h^2 + 5)}{h} = 3x^2 + 3xh + h^2 + 5\].

Key Concepts

Binomial Theorem

The Binomial Theorem is a vital tool in algebra, often used to expand powers of binomials. A binomial is simply a two-term expression like \((x + h)\). The Binomial Theorem provides a formula:\[(x + h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^k\]In simpler terms, this means that each term in the expansion of \((x + h)^3\) involves coefficients derived from Pascal's Triangle.

• The first term is \(x^3\), meaning \(k=0\).
• The second term is \(3x^2h\), calculated as \(\binom{3}{1}x^2h\).
• The third term is \(3xh^2\), calculated as \(\binom{3}{2}xh^2\).
• The final term is \(h^3\), meaning \(k=3\).

This structured approach brings clarity to expanding powers, making lengthy algebraic manipulations simpler. Understanding this theorem is the key to easily expanding binomials into multiple terms.

Factoring Expressions

Factoring is the process of breaking down an expression into simpler components that, when multiplied together, give the original expression. This step is crucial for simplification and solving equations.In our problem, we simplify the expression by factoring out common terms:

• We initially have \(3x^2h + 3xh^2 + h^3 + 5h\).
• Notice each term has an \(h\), so we factor it out, leading to \(h(3x^2 + 3xh + h^2 + 5)\).

By recognizing common factors, the expression becomes more manageable, allowing for the next steps of simplification.

Canceling Terms

Canceling terms involves simplifying an expression by removing factors common to both the numerator and the denominator. But it's important to note that terms can only be canceled when they appear exactly in both components. In the solution, after factoring \(h\) from the numerator, the expression becomes \(\frac{h(3x^2 + 3xh + h^2 + 5)}{h}\).
Given \(heq 0\), the \(h\) in both the numerator and the denominator cancels out:

• Leaving us with the simplified expression: \(3x^2 + 3xh + h^2 + 5\).

This step ensures the expression is as simple as possible, retaining only the terms necessary to describe its mathematical relationship, and is essential for clearly presenting solutions.