Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{\frac{x^{2}+7 x+12}{x^{2}-7 x+12}}{\frac{x^{2}+x-12}{x^{2}-x-12}} $$

Short answer

1

Step-by-step solution

Factor all polynomials

Factor the numerators and denominators of each fraction individually.For the numerator of the first fraction, factorize: \(x^2 + 7x + 12 = (x + 3)(x + 4)\).For the denominator of the first fraction, factorize: \(x^2 - 7x + 12 = (x - 3)(x - 4)\).For the numerator of the second fraction, factorize: \(x^2 + x - 12 = (x - 3)(x + 4)\).For the denominator of the second fraction, factorize: \(x^2 - x - 12 = (x - 4)(x + 3)\).

Rewrite the expression

Substitute the factored forms back into the expression:\[\frac{\frac{(x + 3)(x + 4)}{(x - 3)(x - 4)}}{\frac{(x - 3)(x + 4)}{(x - 4)(x + 3)}}\].

Apply the rule for dividing fractions

Recall that \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}\). Apply this rule to rewrite the expression as a multiplication:\[\frac{(x + 3)(x + 4)}{(x - 3)(x - 4)} \cdot \frac{(x - 4)(x + 3)}{(x - 3)(x + 4)}\].

Simplify the expression

Cancel out common factors in the numerator and denominator:\[ \frac{(x + 3)(x + 4)}{(x - 3)(x - 4)} \cdot \frac{(x - 4)(x + 3)}{(x - 3)(x + 4)} \implies \frac{(x + 3)(x + 4)}{(x - 3)(x - 4)} \cdot \frac{(x - 4)(x + 3)}{(x - 3)(x + 4)} = 1 \].

Key concepts

Factoring Polynomials

When working with rational expressions, one of the first and most crucial steps is to factor polynomials. Factoring helps break down complex expressions into simpler ones.

To factor a polynomial like \(x^2 + 7x + 12\), we need to find two numbers that multiply to the constant term (12) and add to the coefficient of the linear term (7). In this case, the numbers are 3 and 4 because \(3 \times 4 = 12\) and \(3 + 4 = 7\). So, \(x^2 + 7x + 12\) factors to \((x + 3)(x + 4)\).

Here's how we factored each part in our original exercise:

• The numerator of the first fraction: \( x^2 + 7x + 12 = (x + 3)(x + 4) \)
• The denominator of the first fraction: \( x^2 - 7x + 12 = (x - 3)(x - 4) \)
• The numerator of the second fraction: \(x^2 + x - 12 = (x - 3)(x + 4)\)
• The denominator of the second fraction: \( x^2 - x - 12 = (x - 4)(x + 3)\)

Simplifying Rational Expressions

Once we factor the polynomials, the next step is to simplify the rational expressions. This involves rewriting the expressions and looking for common factors that we can cancel out.
In our sample exercise, we first rewrite the given expression using the factored forms:
\[\frac{\frac{(x + 3)(x + 4)}{(x - 3)(x - 4)}}{\frac{(x - 3)(x + 4)}{(x - 4)(x + 3)}}\]
Next, we apply the rule of dividing fractions, which states:
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \]
Therefore, we have:
\[ \frac{(x + 3)(x + 4)}{(x - 3)(x - 4)} \cdot \frac{(x - 4)(x + 3)}{(x - 3)(x + 4)}\]
The expression is now in a form we can easily simplify.

Dividing Fractions and Simplifying

The final step is to simplify the expression even further. Here, we cancel out the common factors in the numerator and denominator.
In our example, the common factors \((x + 3)\), \((x + 4)\), and \((x - 4)\) appear both in the numerator and the denominator. Thus, the simplified form is:
\[ \frac{(x + 3)(x + 4)}{(x - 3)(x - 4)} \cdot \frac{(x - 4)(x + 3)}{(x - 3)(x + 4)} = 1 \]
By canceling out these common terms, we're left with 1, showing that the rational expression simplifies to 1.
This demonstrates the power of factoring in simplifying complex rational expressions!