Problem 79 - Algebra A Combined Function

Question

In your own words, explain how you multiply rational expressions.

Step-by-Step Solution

Answer

Multiply numerators together, multiply denominators, simplify by canceling common factors, and write the result.

Step 1: Understand Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, \( \frac{x^2 - 1}{x + 3} \) is a rational expression.

Step 2: Identify the Expressions to Multiply

Let's say we need to multiply two rational expressions, \( \frac{x^2 - 1}{x + 3} \) and \( \frac{x + 5}{x^2 - 4} \). It's important to understand that each part of these expressions can have variables and constants.

Step 3: Factor All Numerators and Denominators

Before multiplying, factor both the numerators and denominators wherever possible. For example, the polynomial in the numerator \( x^2 - 1 \) factors to \( (x - 1)(x + 1) \) and the denominator \( x^2 - 4 \) factors to \( (x - 2)(x + 2) \).

Step 4: Multiply Numerators Together

Multiply the numerators of the two rational expressions as if they were regular polynomials. Using our example: \( (x^2 - 1)(x + 5) \). After factoring: \( (x - 1)(x + 1)(x + 5) \).

Step 5: Multiply Denominators Together

Multiply the denominators of the two rational expressions. In the example, \( (x + 3)(x^2 - 4) \). After factoring: \( (x + 3)(x - 2)(x + 2) \).

Step 6: Simplify the Expression

Look for any common factors in the numerator and denominator and cancel them out. If there were any cancellations possible, do them now to simplify the expression. For this example, there are no common factors, so no cancellation is possible.

Step 7: Write the Final Expression

Write the simplified expression, if any factors were cancelled out earlier. Final form for our example remains: \( \frac{(x - 1)(x + 1)(x + 5)}{(x + 3)(x - 2)(x + 2)} \).

Key Concepts

Understanding Rational Expressions

Rational expressions are essentially fractions where both the top (numerator) and bottom (denominator) are made up of polynomials. Think of a rational expression as a more complex fraction involving variables. Here's a simple example: \( \frac{x^2 - 1}{x + 3} \). This expression is rational because both \( x^2 - 1 \) and \( x + 3 \) are polynomials.

When dealing with rational expressions, always check that the denominator isn't zero. This means solving for when the denominator equals zero since that value will not be included in the expression's domain. For \( \frac{x^2 - 1}{x + 3} \), make sure \( x + 3 eq 0 \) or, more simply, \( x eq -3 \).

Rationals are powerful because they can represent complex divisions in algebra, similar to how fractions show divisions in arithmetic. They allow us to simplify, multiply, or even divide more intricate expressions, giving us much-needed flexibility in algebraic problem-solving.

Factoring Polynomials

Factoring polynomials is crucial when you want to multiply or simplify rational expressions. Factoring transforms a polynomial into a product of simpler polynomials that, when multiplied together, give you back the original polynomial. For example, the polynomial \( x^2 - 1 \) can be factored into \( (x - 1)(x + 1) \).

Why is factoring important? First, it allows for easier multiplication of polynomials. Second, it helps in identifying common factors within rational expressions, which can then be cancelled to simplify the result. When multiplying rational expressions, always aim to factor both the numerator and the denominator first.

• Identify polynomials: Look for polynomials in both the numerators and denominators.
• Factor completely: Break down each polynomial into its simplest factors.

Once factored, you can proceed to multiply the factored forms. Factoring might seem tricky at first but practice makes it much more intuitive.

Simplifying Expressions

Simplifying expressions, especially after multiplication, involves cancelling out any common factors that appear in both the numerator and the denominator. This step is vital in reducing expressions to their simplest form.

To simplify properly, you need to:

• Identify common factors: After factoring, see if any factors in the numerator are identical to those in the denominator.
• Cancel common factors: If a factor appears in both places, you can cancel them out, much like reducing fractions.
• Re-evaluate: After cancellation, re-evaluate the expression to see if further simplification is possible.

Keep in mind that cancelling is only allowed for product terms, not sums or differences. Simplification ensures the expression is as concise as possible, getting you closer to the solution efficiently. With our example, after multiplying and factoring, you'd test for any common factors in \( \frac{(x - 1)(x + 1)(x + 5)}{(x + 3)(x - 2)(x + 2)} \) and find none to cancel. Always remember: simplification does not change the value of the expression, only its appearance.