Problem 50 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{-6 x^{3}-21 x^{2}+12 x}{-18 x^{3}-42 x^{2}+120 x}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2x^2 + 7x - 4}{2(3x^2 + 7x - 20)}\).

Step 1: Factor the Numerator

First, identify any common factors in the numerator of the expression \(-6x^3 - 21x^2 + 12x\). The Greatest Common Factor (GCF) is \,\ -3x \,\. Factoring \(-3x\) out, we get \(-3x(2x^2 + 7x - 4)\).

Step 2: Factor the Denominator

Next, factor the denominator \(-18x^3 - 42x^2 + 120x\). The GCF here is \,\ -6x \,\. Factoring \(-6x\) out, we obtain \(-6x(3x^2 + 7x - 20)\).

Step 3: Look for a Common Binomial

Upon initial inspection, check if the factored forms can be simplified by cancelling out the common binomial terms in both the numerator and the denominator. However, here we find that there is no common binomial factor. Thus we will only simplify by cancelling out the remaining factor.

Step 4: Simplify by Cancelling Common Terms

Since we factored out the GCFs previously, we can cancel the common term in the numerator and denominator, which is \(-3x\). After cancelling, the expression becomes \(\frac{2x^2 + 7x - 4}{2(3x^2 + 7x - 20)}\).

Step 5: Further Simplify if Possible

Check if further simplification is possible by factoring both quadratic expressions \(2x^2 + 7x - 4\) and \(3x^2 + 7x - 20\). Determine if these expressions can be factored further; however, they are already in simplest forms with no common factors.

Key Concepts

Factoring Polynomials

Factoring polynomials is simplifying an expression by breaking it down into its product of simpler terms. This process is crucial when dealing with rational expressions, which involve a numerator and a denominator, both of which are polynomial expressions. Polynomials can be factored in various ways, such as:

• Finding the greatest common factor (GCF): This is the largest factor that divides each term in the polynomial, and it simplifies the expression significantly.
• Recognizing special polynomial forms: Such as perfect squares or the difference of squares.

When we factor the numerator and the denominator in rational expressions, as shown in the exercise, we can often simplify the expression further by identifying and removing common factors. In our example, the factored form of the numerator is \(-3x(2x^2 + 7x - 4)\) and the denominator is \(-6x(3x^2 + 7x - 20)\). This indicates that the initial step of factoring is effectively setting the stage for further simplification.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that is shared by all the terms in a polynomial. Identifying the GCF is the first and often simplest step in factoring as it can significantly reduce the complexity of the expression.

• Example in the Exercise: For the expression \(-6x^3 - 21x^2 + 12x\), the GCF is \(-3x\). This reduces the polynomial to \(-3x(2x^2 + 7x - 4)\). In the denominator \(-18x^3 - 42x^2 + 120x\), the GCF is \(-6x\), simplifying it to \(-6x(3x^2 + 7x - 20)\).

By factoring out the GCF, we achieve a form that is easier to manipulate and further analyze. It is a powerful technique not just in simplifying rational expressions but also in solving polynomial equations.

Cancelling Common Factors

Cancelling common factors is the subsequent step after factoring the numerator and the denominator of a rational expression. This process is all about simplifying the expression to its most reduced form by eliminating identical factors that appear in both the numerator and the denominator.

• Importance: This step is crucial as it makes the expression easier to work with and understand, especially if further mathematical operations are needed.
• Example in the Exercise: After factoring the numerator \(-3x(2x^2 + 7x - 4)\) and the denominator \(-6x(3x^2 + 7x - 20)\), the common factor \(-3x\) can be cancelled, simplifying the expression to \(\frac{2x^2 + 7x - 4}{2(3x^2 + 7x - 20)}\).

It's important to only cancel factors, not terms; this distinction helps in maintaining the correct mathematical validity of the expression. Simplifying rational expressions by cancelling common factors is a fundamental skill in algebra that aids in expanding one's capacity to solve more complex equations and mathematical models.