Problem 15 - Algebra A Combined Function

Question

Find each product and simplify if possible. See Examples 1 through 3. $$ \frac{x^{2}+6 x+8}{x^{2}+x-20} \cdot \frac{x^{2}+2 x-15}{x^{2}+8 x+16} $$

Step-by-Step Solution

Answer

The simplified product is \( \frac{x^2 - x - 6}{x^2 - 16} \).

Step 1: Factor the Numerators and Denominators

First, we look at the quadratic expressions in both numerators and denominators to see if we can factor them. Start with \( x^2 + 6x + 8 \): it factors to \((x + 2)(x + 4)\). Next, \( x^2 + x - 20 \): it factors to \((x + 5)(x - 4)\). Then, \( x^2 + 2x - 15 \): it factors to \((x + 5)(x - 3)\). Finally, \( x^2 + 8x + 16 \): it factors to \((x + 4)(x + 4)\).

Step 2: Set Up the Expression with Factored Forms

Replace each polynomial in the expression with its factored form. The expression becomes:\[ \frac{(x + 2)(x + 4)}{(x + 5)(x - 4)} \cdot \frac{(x + 5)(x - 3)}{(x + 4)(x + 4)} \]

Step 3: Cancel Common Factors

Examine the fractions for any common factors that can be canceled. The common factors in the numerator and denominator are \(x + 4\) and \(x + 5\). After canceling, the expression simplifies to:\[ \frac{(x + 2)(x - 3)}{(x - 4)(x + 4)} \]

Step 4: Write the Final Simplified Expression

The remaining expression is the simplest form since no further factoring or canceling can be done. The simplified product is:\[ \frac{x^2 - x - 6}{x^2 - 16} \]

Key Concepts

Factoring Quadratics

Factoring quadratics is the backbone of simplifying algebraic fractions. It involves rewriting a quadratic expression in the form of ax^2 + bx + c as a product of two binomials. For example, with the expression \( x^2 + 6x + 8 \), our goal is to find two numbers that multiply to 8 (the constant term) and add to 6 (the linear coefficient). The numbers 2 and 4 satisfy these conditions, so we can factor it as \((x + 2)(x + 4)\).

This process requires identifying patterns or using techniques like the "trial-and-error" method or the "factoring by grouping" method. Often, it's useful to remember the special forms such as the difference of squares \( (a-b)(a+b) = a^2 - b^2 \) and perfect square trinomials \( a^2 ± 2ab + b^2 = (a ± b)(a ± b) \).

Practicing this skill aids in simplifying expressions and solving equations effectively. Ensuring all quadratics are fully factored often reveals common terms that you can later cancel out to simplify the expression further.

Simplifying Expressions

Simplifying expressions involves reducing them so they become easier to work with or interpret. After factoring, the next step is to look for common terms in numerators and denominators that can be canceled. In our given fraction multiplication problem, you'll notice that each fraction is independently factored first. This allows us to see factors such as \(x + 5\) and \(x + 4\) appear in both the numerator and denominator across the fractions.

Canceling these common terms simplifies the expression significantly. It's important to ensure all common factors are removed. This simplification helps in situations requiring further operations or limit calculations.

Keep in mind, you can only cancel factors that are identical and are not part of an addition/subtraction operation within a polynomial term in either the numerator or denominator. After careful simplification, expressions often showcase their simplest structure, like the quadratic \( \frac{(x + 2)(x - 3)}{(x - 4)(x + 4)} \) becoming \( \frac{x^2 - x - 6}{x^2 - 16} \).

Polynomial Multiplication

Polynomial multiplication occurs when you deal with products of polynomials such as when expressions are enclosed in brackets and multiplied. It is an essential method with which students work when simplifying algebraic expressions or solving quadratic problems. Take, for instance, multiplying \((x + 2)(x - 3)\). You apply the distributive property: first each term in the first binomial multiplies each term in the second.

• Multiply the first terms: \(x * x = x^2\)
• Multiply the outer terms: \(x * -3 = -3x\)
• Multiply the inner terms: \(2 * x = 2x\)
• Multiply the last terms: \(2 * -3 = -6\)

Adding these results gives you the polynomial \(x^2 - 3x + 2x - 6\), which simplifies to \(x^2 - x - 6\). This straightforward method ensures students can consistently handle more complex expressions. It reinforces understanding of both the distributive property and the process of combining like terms, two fundamental algebraic skills. Understanding these operations fosters confidence in manipulating and simplifying algebraic fractions.