Problem 62 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{2 c^{3}-2 c^{2}-4 c}{4 c^{3}-8 c^{2}-12 c} $$

Step-by-Step Solution

Answer

\(\frac{c-2}{2(c-3)}\)

Step 1: Factor the Numerator

Identify the common factor in the numerator. The expression in the numerator is \(2c^3 - 2c^2 - 4c\). The common factor is 2c. So, factor out 2c: \(2c(c^2 - c - 2)\).

Step 2: Factor the Denominator

Identify the common factor in the denominator. The expression in the denominator is \(4c^3 - 8c^2 - 12c\). The common factor is 4c. So, factor out 4c: \(4c(c^2 - 2c - 3)\).

Step 3: Factor the Quadratics

Factor the quadratic expressions further. In the numerator: \(c^2 - c - 2 = (c - 2)(c + 1)\). In the denominator: \(c^2 - 2c - 3 = (c - 3)(c + 1)\).

Step 4: Simplify the Fraction

Substitute the factored forms back into the fraction and simplify: \(\frac{2c(c-2)(c+1)}{4c(c-3)(c+1)}\). Cancel out common factors. The common factors are 2c and (c+1), so the simplified fraction is \(\frac{1(c-2)}{2(c-3)} = \frac{c-2}{2(c-3)}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a key skill for simplifying algebraic fractions. The idea is to express a polynomial as a product of simpler polynomials. For example, consider the polynomial 2c³ - 2c² - 4c from our problem. Here, we notice that each term contains a common factor. By factoring out the greatest common factor (GCF), we can break down the polynomial into simpler parts. We pulled out 2c from each term, transforming it into 2c(c² - c - 2).
This method simplifies our work substantially and prepares the polynomial for further simplification steps.

Common Factors

The concept of common factors is crucial when simplifying algebraic expressions. A common factor is a number or variable that divides exactly into each term of an expression. In our problem, both the numerator and the denominator have common factors. In the numerator 2c³ - 2c² - 4c, the common factor is 2c, and in the denominator 4c³ - 8c² - 12c, it is 4c.

• Identifying common factors makes it easier to factor the expressions.
• Once common factors are factored out, we can cancel them out if they appear in both the numerator and denominator.

Spotting common factors early on saves time and helps to avoid errors.

Simplification Steps

Simplifying algebraic fractions involves a series of systematic steps:

Identify & Factor Common Factors: Start by identifying common factors in both the numerator and the denominator. Factor them out to simplify the expressions. In our problem, we identified 2c and 4c as common factors in the numerator and denominator, respectively, and factored them out.

Factor Quadratics: In some cases, like ours, you might need to further factor quadratic expressions. The numerator c² - c - 2 was factored into (c - 2)(c + 1) and the denominator c² - 2c - 3 into (c - 3)(c + 1).

Cancel Common Terms: Substitute the factored forms back into the fraction and cancel out any common terms present in both the numerator and the denominator. This step leaves you with a fully simplified fraction.
By following these steps methodically, you ensure that no crucial elements are missed.

Quadratic Expressions

Quadratic expressions play a significant role in algebra and often appear in problems requiring simplification. A quadratic expression is a polynomial of degree 2, generally in the form ax² + bx + c. In our exercise, both the numerator and the denominator contain quadratic expressions:

• Numerator: c² - c - 2
• Denominator: c² - 2c - 3

Factoring these quadratics involves finding two binomials whose product matches the quadratic expression. For c² - c - 2, we found it factored as (c - 2)(c + 1). Similarly, c² - 2c - 3 factors into (c - 3)(c + 1).
Mastering the factoring of quadratics is essential for simplifying more complex fractions and solving quadratic equations.
Remember, practice makes perfect in identifying patterns and factors quickly.