Problem 52 - Algebra 2

Question

Simplify each expression. \(\frac{\frac{5 x^{2}-5 x-30}{45-15 x}}{\frac{6+x-x^{2}}{4 x-12}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{3}\).

Step 1: Rewrite the Complex Fraction

The given expression is a complex fraction. Begin by rewriting it as a division of two fractions: \[ \frac{\frac{5x^2 - 5x - 30}{45 - 15x}}{\frac{6 + x - x^2}{4x - 12}} = \frac{5x^2 - 5x - 30}{45 - 15x} \div \frac{6 + x - x^2}{4x - 12} \]

Step 2: Factor the Numerator and Denominator

Factor each part of the fractions where possible. - For the numerator of the first fraction, \(5x^2 - 5x - 30\), factor out the greatest common factor (GCF), which is 5: \[ 5(x^2 - x - 6) \] - For \(x^2 - x - 6\), factor into \((x - 3)(x + 2)\). So, \[ 5(x - 3)(x + 2) \] - For the denominator of the first fraction, \(45 - 15x\), factor out \(-15\): \[ -15(x - 3) \] - For the numerator of the second fraction, \(6 + x - x^2\), rewrite it as \(-x^2 + x + 6\) and factor to \[-(x^2 - x - 6) = -(x - 3)(x + 2)\].- For the denominator of the second fraction, \(4x - 12\), factor out the GCF, which is 4: \[ 4(x - 3) \]

Step 3: Rewrite the Division with Factored Expressions

Replace the original complex fraction with these factored versions: \[ \frac{5(x - 3)(x + 2)}{-15(x - 3)} \div \frac{-(x - 3)(x + 2)}{4(x - 3)} \]

Step 4: Simplify Each Fraction

Cancel the common factors in each fraction:- In the first fraction, \((x - 3)\) cancels out: \[ \frac{5(x + 2)}{-15} \] simplifies to: \[ \frac{x + 2}{-3} \] - In the second fraction, \((x - 3)\) also cancels out:\[ \frac{-(x + 2)}{4} \]

Step 5: Multiply the Simplified Fractions

Multiply the simplified results from each fraction:\[ \frac{x + 2}{-3} \times \frac{4}{-(x + 2)} \]Cancel the \((x + 2)\) terms to get:\[ \frac{4}{3} \]

Step 6: State the Simplified Expression

After simplifying all the terms, the final expression is: \[ \frac{4}{3} \]

Key Concepts

Complex Fractions

Understanding complex fractions is an essential skill when dealing with algebraic expressions. In simple terms, a complex fraction is a fraction where the numerator, denominator, or both are themselves fractions. Imagine having a fraction within a fraction! This can sometimes look intimidating, but by systematically simplifying each component, complex fractions become manageable.

Consider this: the first step to simplify a complex fraction, like the one from the exercise, is to rewrite it as a division problem. This transforms one fraction over another into a division between the two. For example:

• The complex fraction \(\frac{\frac{a}{b}}{\frac{c}{d}}\) is equivalent to \(\frac{a}{b} \div \frac{c}{d}\).

This becomes the first step towards simplification, allowing you to work with familiar division operations, converting the whole expression into a more user-friendly form by multiplying numerator and denominator accurately.

Factoring Polynomials

Factoring polynomials is at the heart of simplifying many algebraic expressions, particularly those involving fractions. It refers to the process of breaking down a polynomial into simpler, multiplicative components called factors.

Look at the polynomials given in our exercise. Begin with a polynomial like \(5x^2 - 5x - 30\). One effective way is to first identify the greatest common factor (GCF). Here, the GCF is 5, which can be factored out, simplifying the polynomial to \(5(x^2 - x - 6)\).

Next, you further need to break down the quadratic expression \(x^2 - x - 6\) by finding two numbers that multiply to -6 and add up to -1. This yields \((x - 3)(x + 2)\), a perfect example of how factoring reveals simpler components hidden within polynomials. Such techniques are key to moving further towards simplifying algebraic fractions effectively, as it allows reduction by cancellation of common factors across numerators and denominators.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Just like regular fractions, these expressions can often be simplified by factoring and canceling common factors.

To simplify rational expressions, you first need to identify any factors common to both the numerator and the denominator. In our exercise, rational expressions are found in each part of the complex fraction. By factoring these parts thoroughly, you expose opportunities to cancel equivalent terms, allowing the expression to be reduced.

• This is akin to simplifying the fraction \( \frac{6}{9} \) by recognizing both 6 and 9 share a factor of 3, yielding the simplest form \(\frac{2}{3}\).

When working with rational expressions, always check that the variable restrictions are clear and not violated by the simplification. Remember, a factor resulting in division by zero is invalid and must be excluded. Understanding these little nuances enables mastery over rational expressions and ensures accuracy in your work.