Problem 77 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{3 x^{2}+3 x}{x^{2}-5 x+6}-\frac{3 x^{2}-3 x+12}{x^{2}-5 x+6}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{6}{x-3}\).

Step 1: Identify a Common Denominator

Notice that both fractions share the same denominator: \(x^2 - 5x + 6\). This simplifies our task since we don't need to find a new common denominator.

Step 2: Combine the Numerators

Since the denominators are identical, subtract the numerators of the fractions directly: \( (3x^2 + 3x) - (3x^2 - 3x + 12) \).

Step 3: Simplify the Numerator

Remove parentheses and combine like terms: \(3x^2 + 3x - 3x^2 + 3x - 12\). This reduces to \(6x - 12\).

Step 4: Factor the Numerator

Factor the expression \(6x - 12\). Both terms share a common factor of 6, so factor it out: \(6(x - 2)\).

Step 5: Factor the Denominator

Rewrite the denominator \(x^2 - 5x + 6\) in its factored form. Solve for zeros of the quadratic equation to factor: \((x - 3)(x - 2)\).

Step 6: Simplify the Fraction

The fraction \(\frac{6(x - 2)}{(x - 3)(x - 2)}\) can now be simplified by canceling the \(x - 2\) term from numerator and denominator, resulting in \(\frac{6}{x - 3}\).

Key Concepts

Fractions

Fractions are parts of a whole and they express quantities that are less than one. A fraction consists of two numbers: the numerator (top number) and the denominator (bottom number). In the context of algebra, fractions can also include algebraic expressions, where variables appear in either the numerator, the denominator, or both.

Handling fractions in algebra often requires:

• Finding a common denominator when adding or subtracting them.
• Keeping the denominator the same to directly combine numerators if denominators are identical.

In the example you've seen, the task becomes straightforward because both fractions have the same denominator, allowing you to directly subtract the numerators. This common denominator aspect simplifies the problem and allows for effective manipulation of algebraic equations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means combining like terms, reducing fractions, and clearing up any unnecessary components.

The goal is to express the algebraic expression as cleanly as possible, which makes solving equations easier and results in a more understandable form. Here are key steps:

• Combine like terms by adding or subtracting coefficients of the same variable terms.
• Remove parentheses by distributing any multipliers and simplifying.
• Reduce fractions by canceling common factors.

In the solution walkthrough, simplifying the combined numerator was crucial. After combining like terms, we achieved a reduced expression: from \((3x^2 + 3x) - (3x^2 - 3x + 12)\) to \(6x - 12\). This was done by removing the parentheses, subtracting appropriately, and combining the like terms.

Factoring Polynomials

Factoring polynomials is a method for breaking down a polynomial into simpler components, called factors. This process reverses polynomial multiplication. Factoring is useful for solving equations, simplifying expressions, and finding polynomial roots.

The process usually involves:

• Finding the greatest common factor (GCF) and factoring it out.
• Looking for recognizable patterns, like difference of squares or trinomials that can be factored further.
• Rewriting the polynomial in the form of a product of factors.

In the exercise's step-by-step solution, factoring the numerator \(6x - 12\) first by extracting the GCF (6), and then the denominator \(x^2 - 5x + 6\) into \((x - 3)(x - 2)\) allowed for canceling common factors. This method helped in reducing the fraction to a simpler form, ultimately leading to \(\frac{6}{x - 3}\). By mastering the factoring of polynomials, students can simplify complex expressions and solve equations with ease.