Problem 27 - Elementary Algebra

Question

For exercises 7-32, simplify. $$ \frac{2 x^{2}-5 x-3}{x^{2}-12 x+27} \cdot \frac{x^{2}-15 x+54}{2 x^{2}+13 x+6} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{x + 6} \).

Step 1 - Factor the Numerators and Denominators

First, factor each numerator and each denominator completely. For the first fraction, factor the numerator and denominator: Numerator: \( 2x^2 - 5x - 3 = (2x + 1)(x - 3) \) Denominator: \( x^2 - 12x + 27 = (x - 3)(x - 9) \) For the second fraction, factor the numerator and denominator: Numerator: \( x^2 - 15x + 54 = (x - 6)(x - 9) \) Denominator: \( 2x^2 + 13x + 6 = (2x + 1)(x + 6) \)

Step 2 - Substitute the Factored Forms

Substitute the factored forms back into the original expression: \( \frac{(2x + 1)(x - 3)}{(x - 3)(x - 9)} \cdot \frac{(x - 6)(x - 9)}{(2x + 1)(x + 6)} \)

Step 3 - Cancel Common Factors

Cancel the common factors that appear in both the numerator and the denominator: \( \frac{\cancel{(2x + 1)} \cancel{(x - 3)}}{\cancel{(x - 3)} \cancel{(x - 9)}} \cdot \frac{\cancel{(x - 6)} \cancel{(x - 9)}}{\cancel{(2x + 1)} (x + 6)} \) After cancellation, the expression simplifies to: \( \frac{1}{x + 6} \)

Key Concepts

Factoring Polynomials

Polynomials are algebraic expressions made up of variables and coefficients. When simplifying fractions involving polynomials, the first step is to factor the polynomials in both the numerators and the denominators. Factoring is the process of expressing a polynomial as a product of its simplest factors.
For example, consider the polynomial numerator of the first fraction: \(2x^2 - 5x - 3\).
This can be factored into \((2x + 1)(x - 3)\). Similarly, the denominator \(x^2 - 12x + 27\) can be factored into \((x - 3)(x - 9)\).
Factoring helps us simplify complex expressions by breaking them into simpler parts. Once the polynomials are factored, the expression becomes easier to work with and allows for the next step of simplification.

Canceling Common Factors

Canceling common factors is crucial when simplifying algebraic fractions. After factoring the polynomials, identify factors that are common in both the numerator and the denominator. These common factors can be 'canceled', or removed, because any number or expression divided by itself is equal to one.
In the simplified expression:

\(\frac{(2x + 1)(x - 3)}{(x - 3)(x - 9)} \cdot \frac{(x - 6)(x - 9)}{(2x + 1)(x + 6)}\)

Common factors such as \((2x + 1)\), \((x - 3)\), and \((x - 9)\) can be canceled:

\(\frac{\cancel{(2x + 1)} \cancel{(x - 3)}} {\cancel{(x - 3)} \cancel{(x - 9)}} \cdot \frac{\cancel{(x - 6)} \cancel{(x - 9)}} {\cancel{(2x + 1)} (x + 6)}\)

After canceling, we are left with \(\frac{1} {x + 6}\).
Canceling simplifies the expression and makes it easier to understand and solve.

Algebraic Fractions

An algebraic fraction is a fraction where both the numerator and the denominator are algebraic expressions, typically involving variables. Simplifying these fractions involves factoring the polynomials and canceling out common factors. It's essential to first factor the numerator and denominator completely.
Let's review the initial given expression:

\(\frac{2x^2 - 5x - 3} {x^2 - 12x + 27} \cdot \frac{x^2 - 15x + 54} {2x^2 + 13x + 6}\)

After factoring, we substitute factored forms:

\(\frac{(2x + 1)(x - 3)} { (x - 3)(x - 9) } \cdot \frac{(x - 6)(x - 9)} { (2x + 1)(x + 6) }\)

Canceling the common factors results in:
\(\frac{1} {x + 6}\)
Understanding how to simplify algebraic fractions is essential for tackling more complex algebra problems. It also builds a solid foundation for future topics, such as rational expressions and polynomial equations.