Problem 19 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find the product. $$ \frac{x^2+5 x-36}{x^2-49} $$ \(\left(x^2-11 x+28\right)\)

Step-by-Step Solution

Answer

The product of the given expressions is \( \frac{x+9}{x+7} \).

Step 1: Factorize the expressions

First, factorize each expression. The rational expression \( \frac{x^2+5 x-36}{x^2-49} \) can be factored as \( \frac{(x-4)(x+9)}{(x-7)(x+7)} \) \n The expression \( x^2-11 x+28\) is a quadratic expression, it can be factored as \( (x-7)(x-4) \)

Step 2: Multiply factorized expressions

Next, multiply the factorized expressions \( \frac{(x-4)(x+9)}{(x-7)(x+7)} \) and \( (x-7)(x-4) \) together.

Step 3: Simplify the product

The (x-4) and (x-7) terms will cancel out from the numerator and denominator to give the final product as \( \frac{x+9}{x+7} \)

Key Concepts

Factoring Quadratics

Factoring quadratics is a key skill in algebra that helps simplify expressions and solve equations. A quadratic expression typically takes the form \( ax^2 + bx + c \). The goal is to rewrite this expression as a product of two binomials. Here are a few useful steps to understand the factoring process better:

• Identify coefficients: Look at the coefficients of the quadratic term (\(a\)), the linear term (\(b\)), and the constant term (\(c\)).
• Multiply \(a\) and \(c\): Find two numbers that multiply to \(a \times c\) and add up to \(b\).
• Rewrite and factor: Use these numbers to split the middle term, rewrite the expression, and then factor by grouping.

For example, the expression \(x^2 - 11x + 28\) is factored by finding two numbers that multiply to 28 and add up to -11, which are -7 and -4. Thus, it factors to \((x-7)(x-4)\).
Mastering factoring is crucial for simplifying and solving more complex algebraic expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often includes factoring, but also entails cancelling out terms that appear in both the numerator and the denominator of a rational expression. Here are the steps to simplify effectively:

• Factor first: Ensure that each part of the expression is factored completely.
• Cancel common terms: Look for terms that appear in both the numerator and the denominator and cancel them.
• Re-evaluate: Check the remaining terms and ensure the expression is as simple as possible.

For example, after factoring \(\frac{x^2 + 5x - 36}{x^2 - 49}\) into \(\frac{(x-4)(x+9)}{(x-7)(x+7)}\), and multiplying it by \((x-7)(x-4)\), we cancel the \((x-4)\) and \((x-7)\) terms to arrive at the simplified expression \(\frac{x+9}{x+7}\).
Simplifying expressions is important because it makes calculations easier and helps in understanding the underlying mathematical relationships.

Multiplying Rational Expressions

Multiplying rational expressions involves multiplying the numerators together and the denominators together. Afterward, it's crucial to simplify the expression by cancelling common factors. Let's break it down:

• Factor both expressions: Start by factoring all numerators and denominators completely.
• Multiply across: Perform multiplication by multiplying all numerators to create a new numerator and all denominators to form a new denominator.
• Cancel common factors: Once you have the new numerator and denominator, cancel out any common terms that appear in both.

In the given exercise, after factoring, the expression is \(\frac{(x-4)(x+9)}{(x-7)(x+7)}\) multiplied by \((x-7)(x-4)\). By multiplying, we first get \((x-4)(x+9)(x-7)(x-4)\) over \((x-7)(x+7)(x-7)(x-4)\), then cancel the terms \((x-4)\) and \((x-7)\) from both numerator and denominator, resulting in \(\frac{x+9}{x+7}\).
Understanding how to multiply and simplify rational expressions is fundamental for solving more complex algebraic problems efficiently.