Problem 18 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find the product. $$ \frac{x^2-x-6}{4 x^3} \cdot \frac{2 x^2+2 x}{x^2+5 x+6} $$

Step-by-Step Solution

Answer

The product of the given fractions is thus simplified to \( \frac{(x-3)(x+1)}{2x (x+3)} \).

Step 1: Factorize the Quadratic Expressions

Factorize the quadratic expression \(x^2 - x - 6\) in the numerator of the first fraction and the quadratic expression \(x^2 + 5x + 6\) in the denominator of the second fraction in order to express them as product of binomials. The first expression can be factored into \((x-3)(x+2)\) and the second expression into \((x+2)(x+3)\).

Step 2: Express Fractions with Factored Expressions

Substitute the factored expressions into the fractions. The given expression can therefore be written as: \[\frac{(x-3)(x+2)}{4 x^3} \cdot \frac{2 x(x+1)}{(x+2)(x+3)}\]

Step 3: Simplify the Fractions

Simplify the fractions by cancelling out common factors in the numerator and the denominator. The \(x+2\) terms in the numerator of the first fraction and the denominator of the second fraction will cancel each other out. Thus, the simplified product will be:\[\frac{(x-3)2x(x+1)}{4 x^3 (x+3)}\]

Step 4: Further Simplify

To further simplify, factorize \(2x\) in the numerator and \(4x^3\) in the denominator to cancel out common factors:\[\frac{x(x-3)(x+1)}{2 x^2 (x+3)}\]This reduces to:\[\frac{(x-3)(x+1)}{2x (x+3)}\]

Key Concepts

Factoring Quadratics

When you encounter quadratic expressions like \( x^2 - x - 6 \) or \( x^2 + 5x + 6 \), the process of factoring them is essential in many mathematical operations like simplifying and solving equations. Quadratics can often be rewritten as a product of two binomial expressions. Here's how it works:

• Identify two numbers that multiply to give the constant term (the last number) and add to give the middle coefficient (the coefficient of \( x \)).
• For \( x^2 - x - 6 \), these numbers are \(-3\) and \(2\). Factoring results in \( (x-3)(x+2) \).
• For \( x^2 + 5x + 6 \), the numbers \(2\) and \(3\) are used, leading to \((x+2)(x+3)\).

Once factored, expressions become easier to manipulate, often allowing for cancellation or other simplifications in broader equations.

Simplifying Fractions

Simplifying fractions in algebra involves canceling common terms in the numerator and the denominator. Here's a step-by-step approach:

• Ensure all parts of the fractions are factored completely. This means breaking down all polynomial expressions into their simplest binomial or monomial components.


• Identify any common factors in both the numerator and the denominator that can be canceled out. In the product \( \frac{(x-3)(x+2)}{4 x^3} \cdot \frac{2 x(x+1)}{(x+2)(x+3)} \), the \( x+2 \) terms cancel each other out.


• Simplification often results in a more manageable expression, reducing complexity and making further operations, like multiplication or division, easier to perform.

Remember, simplifying fractions doesn't change the value the expression represents; it just makes it more accessible for calculation.

Polynomial Expressions

Polynomial expressions form the backbone of a lot of algebraic operations. Understanding how to manipulate them is crucial:

• A polynomial is simply the sum of terms, each consisting of a coefficient and a variable raised to a power, such as \( 2x^3 + 5x^2 + x + 6 \).
• Operations like addition, subtraction, and multiplication are intrinsic to working with polynomials. Factoring them, as seen in the quadratic expressions, can significantly simplify these operations.


• In the exercise, recognizing polynomials in both the fractions and utilizing their factored forms helps in multiplying and simplifying the entire expression efficiently.

Polynomials often appear complex, but with factoring and simplification, they can be managed and understood with ease.