Problem 43 - Algebra and Trigonometry

Question

\(43-44\) Simplify using the Binomial Theorem. $$ \frac{(x+h)^{3}-x^{3}}{h} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Binomial Theorem

The Binomial Theorem states \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \). In this exercise, apply the binomial theorem to \( (x + h)^3 \).

Step 2: Apply the Binomial Expansion

Expand \( (x + h)^3 \) using the Binomial theorem: \( (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \).

Step 3: Substitute and Simplify

Substitute \( (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \) into the original expression \( \frac{(x+h)^{3}-x^{3}}{h} \). This gives \( \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} \).

Step 4: Cancel Similar Terms

Simplify the expression by canceling \( x^3 - x^3 \), resulting in \( \frac{3x^2h + 3xh^2 + h^3}{h} \).

Step 5: Factor and Simplify

Factor out \( h \) from the numerator: \( \frac{h(3x^2 + 3xh + h^2)}{h} \). Cancel \( h \) from the numerator and denominator, which simplifies to \( 3x^2 + 3xh + h^2 \).

Key Concepts

Polynomial Expansion

The polynomial expansion of a binomial expression involves expressing the expression as a sum of terms. This is achieved using the Binomial Theorem, which provides a formula for expanding a binomial raised to any power. The theorem can be written as \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\). This allows us to expand expressions like \((x + h)^3\) in a systematic way.

• The terms generated are based on the powers of each variable, following the order from the highest power to the lowest.
• Each term is made up of a combination of coefficients given by the binomial coefficients \(\binom{n}{k}\), which can be found in Pascal's triangle.

The polynomial \((x + h)^3\) expands to \(x^3 + 3x^2h + 3xh^2 + h^3\), capturing the interaction of \(x\) and \(h\) in powers that sum to 3. Understanding polynomial expansion is crucial for simplifying expressions efficiently, especially when dealing with higher powers.

Simplification Techniques

Simplifying expressions means reducing them to their simplest or most basic form without altering their value. In many exercises, especially using the Binomial Theorem, simplification involves several key strategies:

• Cancellation: Once expansions occur, identify terms that cancel each other out. This helps reduce the complexity of the expression. For instance, in \(\frac{(x+h)^3-x^3}{h}\), after expanding and simplifying, \(x^3\) terms cancel out.
• Factorization: Look for common factors in the terms. Factorization allows you to simplify through division. In this problem, terms were factored by \(h\), allowing for cancellation in the fraction.
• Reduction of Fractions: When numerators and denominators share factors, cancel them to simplify the fraction. This is the final step after expansion in the original exercise.

Simplification not only makes expressions easier to understand but also prepares them for further mathematical operations. In this case, the expression is narrowed down to \(3x^2 + 3xh + h^2\), making it more manageable for analysis or further calculations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the basis of algebra and are fractionally linked to the concepts of polynomial expansion and simplification. In exercises involving expressions like \((x + h)^3\), you work with an expression that transforms through various algebraic processes:

• Terms and Coefficients: Understanding how terms (with variables raised to exponents) and coefficients (numbers that multiply the terms) interact is key. Each term in \((x+h)^3\) contributes to the overall expression.
• Operations: Operations like addition, subtraction, multiplication, and division are used to manipulate the terms. Identifying the sequence of operations helps maintain the integrity of the expression.
• Structure: Well-structured expressions adhere to algebraic rules, allowing for predictable outcomes when simplifying. This structural understanding assists in reducing complex expressions to simple forms.

In exercises like the given one, practicing the handling of algebraic expressions aids in recognizing patterns, simplifies processing, and prepares you for solving more complex algebraic problems. Mastery of these expressions enables students to engage confidently in algebraic manipulation across various contexts.