Problem 81 - Algebra A Combined Function

Question

Factor each numerator and denominator. Then simplify if possible. $$ \frac{6 a^{2} b-9 a b}{3 a b} $$

Step-by-Step Solution

Answer

The simplified expression is \(2a - 3\).

Step 1: Factor the Numerator

The numerator is \(6a^2b - 9ab\). We can factor out the greatest common factor, which is \(3ab\). Doing so gives us: \[6a^2b - 9ab = 3ab(2a - 3)\].

Step 2: Factor the Denominator

The denominator is \(3ab\). Since it is a monomial, it's already in its factored form.

Step 3: Simplify the Fraction

Substitute the factored form of the numerator and the denominator into the fraction: \(\frac{3ab(2a - 3)}{3ab}\). Since \(3ab\) is a common factor in the numerator and denominator, we can cancel it out: \(\frac{3ab}{3ab} \cdot (2a - 3) = 1 \cdot (2a - 3)\). The simplified form is \(2a - 3\).

Key Concepts

Factoring Polynomials

Polynomials are algebraic expressions that involve variables raised to non-negative integer powers. Factoring them involves breaking them down into simpler terms, or 'factors,' that when multiplied together give the original polynomial.
This technique is crucial in simplifying rational expressions efficiently. Let's consider polynomials like the numerator in our exercise, which is \(6a^2b - 9ab\).
Notice that each term in this polynomial shares common factors. By finding these factors, we can express the polynomial in a simpler form.

• Identify the common factor in each term.
• Extract this common factor outside of a bracket.
• Rewrite the polynomial as the common factor multiplied by a simpler polynomial.

In our example, the greatest common factor is \(3ab\), allowing the numerator to be written as \(3ab(2a - 3)\). This compact form reveals how the polynomial is constructed, assisting with further simplification in expressions.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means canceling out any common factors in the numerator and the denominator. Doing this makes the fraction easier to work with and solve.
In algebra, where expressions often contain polynomials, this simplifies our work significantly. Here's how you can simplify a fraction:

• Factor both the numerator and the denominator into their simplest terms.
• Identify common factors in both parts.
• Cancel the common factors, leaving the fraction in its simplest form.

For instance, take our exercise: we started with \(\frac{6a^2b - 9ab}{3ab}\). By factoring each part: \(\frac{3ab(2a - 3)}{3ab}\), we canceled \(3ab\), simplifying it to \(2a - 3\). This process not only makes the fraction neater but also shows the essence of what it represents.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that two or more numbers or terms share. It's especially handy when simplifying fractions and factoring polynomials.
In algebraic expressions like \(6a^2b - 9ab\), finding the GCF helps in reducing complex terms into more manageable components. Here's the approach:

• List out the factors of each term.
• Identify the largest factor that is common across all terms.
• Use this factor to simplify the expression or polynomial.

For the exercise problem, the GCF of \(6a^2b\) and \(9ab\) is \(3ab\). By factoring this GCF out, it makes simplifying the fraction straightforward, showing again its value in algebraic simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). These expressions form the building blocks of algebra and can be simple like \(3x\) or complex like \((2x^2 + 5x - 3)\).
Understanding them involves recognizing components like terms and coefficients, which simplifies operations like factoring and simplifying. Consider their importance in rational expressions:

• Terms are parts of the expression separated by plus or minus signs.
• Coefficients are numbers multiplying the variables.
• Operations on these expressions should preserve equivalence.

Algebraic expressions are key in representing real-world problems and equations. In our exercise, the expression \(6a^2b - 9ab\) is simplified by addressing each part of it, showcasing how core algebra principles apply.