Problem 70 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the rational expression, if possible. \(\frac{x^3-2 x^2-24 x}{x^2-2 x-24}\)

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(x\).

Step 1: Factor the numerator and the denominator

Let's factor the numerator \(x^3-2 x^2-24 x\) and denominator \(x^2-2 x-24\). This can be done by first taking out the common factors from each expression, which are \(x\) in the numerator and \(1\) in the denominator. \n After factoring, the expression becomes: \(\frac{x(x^2-2 x-24)}{1 * (x^2 - 2x -24)}\).

Step 2: Factor the quadratic

Let's further factor the quadratic \(x^2 - 2x - 24\) in both the numerator and the denominator. The factors of -24 which add up to -2 are -6 and 4. \n The factored form of the quadratic is \((x - 6) * (x + 4)\). The rational expression becomes: \(\frac{x(x - 6)(x + 4)}{(x - 6)(x + 4)}\).

Step 3: Simplify the expression

Now that both the numerator and the denominator are factored, we can see some common factors. The common factors of the numerator and the denominator are \((x - 6)\) and \((x + 4)\). Cancel out these common factors, which leaves us with \(x\) as the simplified form of the given rational expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that helps simplify expressions and solve equations. To factor a polynomial, you look for the greatest common factor (GCF) that can be extracted from all terms in the polynomial. Once identified, you divide each term by this factor.
Consider the numerator from the expression \(x^3 - 2x^2 - 24x\). All terms share the factor \(x\), so we factor it out. This process simplifies the expression to \(x(x^2 - 2x - 24)\).
Some useful tips for factoring polynomials:

• Identify the GCF for all terms.
• Re-write the expression by separating the GCF.
• Check each factor individually to see if it can be factored further.

Quadratic Factoring

Quadratic factoring involves breaking down a quadratic polynomial into a product of two binomials. The general form of a quadratic expression is \(ax^2 + bx + c\). To factor a quadratic expression like \(x^2 - 2x - 24\), find two numbers that multiply to \(c\) (the constant term) and add up to \(b\) (the linear coefficient).
For \(x^2 - 2x - 24\):

• The product \((-6) \cdot 4 = -24\).
• When added together, \((-6) + 4 = -2\).

This gives us the factors \((x - 6)\) and \((x + 4)\).
Quadratic factoring simplifies numerator and denominator into a product of binomials, making simplification possible.

Rational Expression Simplification

Simplifying rational expressions involves reducing the expression to its simplest form by canceling out common factors from the numerator and the denominator. This procedure is crucial as it makes calculations easier and reveals underlying properties.
In our example, after factoring, the expression is \(\frac{x(x - 6)(x + 4)}{(x - 6)(x + 4)}\). Since \((x - 6)\) and \((x + 4)\) appear in both, they can be canceled out. Hence, the expression reduces to just \(x\).
To simplify rational expressions effectively, always:

• Factor both the numerator and the denominator completely.
• Identify and cancel common factors between the numerator and the denominator.
• Ensure no terms are left, and cross-check your work for accuracy.

This process not only simplifies the expression but also helps in finding domains, solving equations, and more.