Problem 80 - Algebra A Combined Function

Question

Factor each numerator and denominator. Then simplify if possible. $$ \frac{x^{3}-8}{4 x-8} $$

Step-by-Step Solution

Answer

\(\frac{x^2 + 2x + 4}{4}\)

Step 1: Factor the Numerator

The numerator is a difference of cubes: \(x^3 - 8\). This can be factored using the formula for the difference of cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = x\) and \(b = 2\). So we have:\[x^3 - 8 = (x - 2)(x^2 + 2x + 4)\]

Step 2: Factor the Denominator

The denominator is a common factor expression. Find the greatest common factor of \(4x - 8\):\[4x - 8 = 4(x - 2)\]

Step 3: Simplify the Expression

Now, place the factored numerator and denominator back into the fraction:\[\frac{(x-2)(x^2 + 2x + 4)}{4(x-2)}\]We can cancel out the common factor \((x - 2)\) from the numerator and the denominator (provided \(x eq 2\) to avoid division by zero):\[\frac{x^2 + 2x + 4}{4}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a polynomial into simpler components called factors. This process allows us to express the polynomial as a product of its factors, which can simplify solving equations or evaluating expressions. When factoring polynomials, it's helpful to look for common patterns or special formulas, such as quadratic trinomials, perfect square trinomials, and the difference of squares or cubes. In our exercise, the polynomial in the numerator, \(x^3 - 8\), is recognized as a difference of cubes. To factor a polynomial, always check if it can be decomposed using known formulas or by finding the greatest common factor (GCF) among its terms. Bringing these components together enables the simplification of algebraic fractions, making calculations easier and expressions more manageable.

Difference of Cubes

The difference of cubes is a special pattern seen in algebra, characterized by the expression \(a^3 - b^3\). There is a specific formula used to factor these expressions, which is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Recognizing this pattern allows us to break down complex cubic expressions into linear and quadratic components. In the given problem, the term \(x^3 - 8\) fits this pattern, as it can be rewritten as \(x^3 - 2^3\). Following the formula, this becomes \((x - 2)(x^2 + 2x + 4)\). By applying the difference of cubes formula, we transform a complex expression into a compact factored form, which is easier to manipulate and simplify in further calculations.

Simplifying Fractions

Simplifying fractions in algebra involves reducing the fraction to its simplest form by eliminating common factors from the numerator and the denominator. This often involves factoring both parts and canceling out common factors. For the expression given in the exercise, after factoring, we have \(\frac{(x-2)(x^2 + 2x + 4)}{4(x-2)}\). Observing the fraction, we notice that \((x-2)\) is a common factor in both the numerator and the denominator.

• Be careful to ensure that cancelling does not violate mathematical rules, like division by zero.
• In this case, \(x\) should not be equal to 2 in the expression \(x-2\), since that would make the original denominators zero.

Once the \((x-2)\) terms are cancelled, the simplified form becomes \(\frac{x^2 + 2x + 4}{4}\). Simplifying fractions makes them easier to work with and is a critical skill in solving more advanced algebraic problems.

Greatest Common Factor

The greatest common factor (GCF) of an expression is the largest factor that divides all terms within the expression. It is a valuable tool used to simplify expressions by factoring out these commonalities. In the problem, we are asked to find the GCF of the denominator \(4x - 8\). By observing each term, \(4x\) and \(-8\), we see they both are divisible by 4. Thus, the GCF is 4, and factoring it out we get \(4(x - 2)\). Recognizing and factoring out the GCF not only simplifies the algebraic expression but also plays a key role in reducing fractions to their simplest forms, significantly aiding in the process of operations like simplification and solving equations.