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Chapter 11: Problem 1

Describe the steps used to multiply two rational expressions.

Short Answer

Expert verified
To multiply two rational expressions, first factorize their numerators and denominators. Cross-cancel any common factors between them before multiplying the numerators together and the denominators together. Finally, simplify the result if necessary.

Step by step solution

01

Understand the Expression

Write down the two rational expressions that are to be multiplied. Each expression will have a numerator and a denominator which are both polynomials.
02

Factorize Numerators and Denominators

Factorize the numerator and the denominator for each of the rational expressions. Expand them out into their simplest polynomial expressions.
03

Cross Cancelation

Look for common factors between the numerators and denominators. These can be cross-cancelled. If factor A is in the numerator and denominator of the expressions, simplify by dividing each by factor A.
04

Perform Multiplication

Practice the actual multiplication. Multiply the remaining factors in the numerator to get the numerator of the resulting expression, and do the same thing for the denominator.
05

Simplify

Check if there are any common factors left between the numerator and denominator. If yes, simplify by dividing each by that common factor. If not, the expression is already simplified. Write down the resulting rational expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorization
Factorization is an essential step when multiplying rational expressions. It involves breaking down each polynomial in the numerators and denominators to its most basic components, the factors. Think of each polynomial as a collection of numbers and variables that can be multiplied together to give the same polynomial. By factorizing these, you are essentially rewriting the expression in its simplest form.
  • Start with the numerator: Look for any common factors or special cases such as the difference of squares, trinomials, or other recognizable patterns.
  • Do the same for the denominator: Once again, search for any factors or special polynomial patterns that apply.
Factorization helps in the later stage of cross-cancellation by making the common factors more evident. It鈥檚 like simplifying a fraction before you further work with it.
Cross Cancellation
Cross cancellation can simplify the multiplication of rational expressions significantly. After factorization, you should look for common factors across the fraction lines. If you find the same factor in both a numerator and a denominator, you can cancel these out. This step is known as cross-cancellation.
  • Check the numerators and denominators: Identify any matching factors.
  • Cancel out these matching factors: Remember, it's vital to only cancel factors that are exactly the same.
  • The idea is to simplify the expression: The fewer factors you have to multiply, the simpler the operation becomes.
By using cross-cancellation, you reduce the complexity and size of numbers involved, making your job much easier. It's one of the tricks that makes working with fractions and rational expressions much more manageable.
Polynomial Simplification
Polynomial simplification comes after you've factored and cross-cancelled your expression. This step ensures that the resulting rational expression is in its simplest possible form. Often after multiplying and cancelling, some factors might still be present in both the numerator and denominator.
  • Look for any remaining common factors: Sometimes not everything gets cancelled out in the cross cancellation step.
  • Divide these factors out: This leaves you with the simplest version.
  • Verify your final result: Make sure both the numerator and denominator are in their simplest forms, with no common factors remaining.
Simplification is necessary because it gives you a concise and neat expression. A simplified polynomial not only looks better but is also easier to handle in further calculations or evaluations.

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