Question

Challenge Problems $$\frac{2 a^{3}+6 a^{2}-8 a}{2 a^{3}+2 a^{2}-4 a}$$

Short answer

\(\frac{a+4}{a+2}\)

Step-by-step solution

Factor out the Greatest Common Factor (GCF) from the numerator and the denominator

Identify and factor out the greatest common factor from the numerator and the denominator. The GCF of the numerator is 2a and the GCF of the denominator is also 2a. Factoring them out, the expression becomes:\[\frac{2a(a^2+3a-4)}{2a(a^2+a-2)}.\]

Factor the quadratic expressions

Factor the quadratic expressions in the numerator and denominator. For the numerator, factors of -4 that add up to 3 are 4 and -1, giving us:\[(a+4)(a-1).\]For the denominator, factors of -2 that add up to 1 are 2 and -1, yielding:\[(a+2)(a-1).\]The factored form then becomes:\[\frac{2a(a+4)(a-1)}{2a(a+2)(a-1)}.\]

Cancel out the common terms

Cancel out the common terms that appear in both the numerator and the denominator. The terms 2a and \((a-1)\) are common and can be canceled out, resulting in:\[\frac{a+4}{a+2}.\]

Key concepts

Greatest Common Factor

Understanding the concept of the greatest common factor (GCF) is essential when simplifying algebraic expressions, especially when dealing with polynomials. The GCF is the largest factor that divides two or more numbers or terms without leaving a remainder. It's the 'greatest' number you can 'factor out' from a set of terms.

In the context of our algebraic fraction, finding the GCF simplifies the process of factoring quadratic expressions. To determine the GCF of an algebraic expression, look for the highest degree of any common variables and the largest number that divides all the coefficients. In our exercise, the GCF of the numerator and the denominator was identified as 2a. By factoring 2a out of both the numerator and the denominator, we set up a simplified structure that can be further factored and reduced.

Algebraic Fraction Simplification

Algebraic fraction simplification is much like simplifying numerical fractions; the goal is to make the expression as simple as possible. A critical step in this process is factoring, similar to breaking down numbers into their prime factors. With algebraic expressions, you break down polynomials into products of their simplest polynomials.

Once the expression is factored, you can often simplify the fraction by canceling out common factors from the numerator and the denominator. Simplification makes the expression more manageable and easier to work with, whether for further operations or for evaluating variable values. In the given exercise, after we factored the quadratic expressions in both the numerator and the denominator, the result was a simplified algebraic fraction that could be readily analyzed and simplified further.

Canceling Common Terms

Canceling common terms is analogous to reducing fractions in basic arithmetic. When we cancel common terms in the numerators and denominators of algebraic fractions, we're essentially removing factors that are identical, thus simplifying the expression. It's a vital step in algebraic fraction simplification, allowing us to get to the simplest form of the fraction.

In our exercise, the terms 2a and \(a-1\) were present in both the numerator and the denominator after factoring. These common terms were canceled out, leaving us with a more straightforward fraction. It's important to remember that you can only cancel terms that are multiplied together, not those that are added or subtracted. This cancellation significantly reduces the complexity of algebraic fractions and leads us to the final simplified form of the expression.