Problem 33 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{2 m^{2}-5 m-12}{m^{2}-10 m+24} \div \frac{4 m^{2}-9}{m^{2}-9 m+18}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite Division as Multiplication

First, let's express the division operation in terms of multiplication. For this, we take the reciprocal of the second fraction. Thus, we rewrite: \[ \frac{2 m^2 - 5 m - 12}{m^2 - 10 m + 24} \times \frac{m^2 - 9 m + 18}{4 m^2 - 9} \]

Step 2: Factor All Polynomials

Next, we need to factor each polynomial in the numerators and denominators:- \(2m^2 - 5m - 12 = (2m + 3)(m - 4)\)- \(m^2 - 10m + 24 = (m - 6)(m - 4)\)- \(4m^2 - 9 = (2m + 3)(2m - 3)\) (difference of squares)- \(m^2 - 9m + 18 = (m - 3)(m - 6)\)

Step 3: Substitute Factored Forms

Substitute the factored forms into the expression:\[ \frac{(2m + 3)(m - 4)}{(m - 6)(m - 4)} \times \frac{(m - 3)(m - 6)}{(2m + 3)(2m - 3)} \]

Step 4: Cancel Common Factors

Cancel the common factors present in the numerators and denominators:- The \((2m + 3)\) in the first numerator and second denominator can be canceled.- The \((m - 4)\) in the first numerator and denominator can be canceled.- The \((m - 6)\) in the second numerator and the first denominator can be canceled.After canceling, we are left with:\[ \frac{1}{1} \times \frac{m - 3}{2m - 3} \]

Step 5: Simplify the Expression

Simplify the remaining expression:\[ \frac{m - 3}{2m - 3} \]This is the simplest form of the expression after simplifying.

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a complex polynomial into simpler, more manageable parts, known as factors. These factors are polynomials of lower degree that, when multiplied together, reproduce the original polynomial. We often use factoring to simplify polynomial expressions and solve polynomial equations.

To factor a polynomial, you can follow these steps:

• Look for a common factor in all terms of the polynomial.
• Check if the polynomial can be factored using special formulas, like the difference of squares or sum/product of roots.
• For quadratics, use techniques like grouping or the quadratic formula to find roots, then express the polynomial as products of its factors.

For example, the polynomial \(4m^2 - 9\) was factored using the difference of squares formula, resulting in \((2m + 3)(2m - 3)\). Similarly, each part of the exercise was broken into binomials, facilitating the next steps in simplifying the expression.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. These expressions can be manipulated much like fractions with numbers, using operations such as addition, subtraction, multiplication, and division.

When working with rational expressions, it's essential to:

• Factor the polynomials in the numerators and denominators to simplify the expression.
• Identify any restrictions by noting the values that make the denominator zero, as they are excluded from the domain.
• Rewrite division problems as multiplication by taking the reciprocal of the divisor.

In the original problem, division was converted into multiplication by using the reciprocal of the divisor. Simplifying rational expressions often involves cancelling common factors, making the expression easier to evaluate or further work with.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This is achieved by combining like terms, factoring, and cancelling common factors. The goal is to make expressions as compact and easy to understand as possible.

To simplify a complex rational expression:

• Factor both the numerator and denominator completely.
• Cancel out any common factors between numerators and denominators.
• Rewrite the expression in the simplest form, ensuring it's easy to interpret and use in calculations.

In the provided exercise, after factoring and cancelling the common factors like \((2m + 3)\), \((m - 4)\), and \((m - 6)\), we ended with the simplified expression \(\frac{m - 3}{2m - 3}\). This process highlights the importance of each simplification step, ensuring the expression is manageable and straightforward.