Multiplication of Rational Functions

Definition

Multiplication of rational functions involves taking two rational expressions and multiplying them to create a new rational expression. Each rational function is expressed as the ratio of two polynomials, and when multiplying, the numerators are multiplied together and the denominators are multiplied together, forming a single fraction. This operation requires attention to simplification and factoring to ensure the resulting expression is in its simplest form.

5 Must Know Facts For Your Next Test

1. When multiplying rational functions, if either function has a common factor in the numerator and denominator, it can be simplified before or after multiplication.
2. The result of multiplying two rational functions will always yield another rational function, provided neither denominator equals zero.
3. The degree of the resulting polynomial in the numerator is the sum of the degrees of the polynomials in the numerators of the original functions.
4. Similarly, the degree of the resulting polynomial in the denominator is the sum of the degrees of the polynomials in the denominators of the original functions.
5. Always check for restrictions on variables after multiplication, as factors that were canceled could introduce new restrictions in the final expression.

Review Questions

• How do you perform multiplication on two rational functions, and what steps must you take to ensure accuracy?
  • To multiply two rational functions, first multiply their numerators together to form a new numerator and their denominators together to create a new denominator. After this, simplify by factoring out any common factors from both the numerator and denominator. It's crucial to check for any restrictions on variable values that might have been overlooked during multiplication. This ensures that the final expression is both accurate and valid.
• Discuss how the degrees of polynomials in rational functions affect the result when they are multiplied together.
  • When multiplying two rational functions, the degree of the resulting polynomial in the numerator equals the sum of the degrees of their respective numerators. Likewise, for the denominator, its degree will be the sum of the degrees from both denominators. This means that if you understand how degrees work in individual polynomials, you can predict how complex your final rational function might be. Keeping an eye on these degrees helps in identifying behavior such as end behavior or asymptotes in graphs.
• Evaluate how multiplication affects potential restrictions on variable values within rational functions and why this matters.
  • Multiplying rational functions can introduce new restrictions on variable values that were not present before. When you cancel common factors during multiplication, it may seem that those factors no longer influence restrictions. However, those canceled factors still dictate values that cannot be included in the final expression. Understanding these restrictions is crucial because they affect the domain of your resulting function. Ignoring these could lead to incorrect conclusions about where your function is defined and how it behaves.

Related terms

Rational Function: A function represented as the ratio of two polynomial functions, where the denominator is not equal to zero.

Polynomial: A mathematical expression consisting of variables raised to whole number powers, combined using addition, subtraction, and multiplication.

Simplification: The process of reducing a mathematical expression to its simplest form by eliminating common factors or combining like terms.