key term - Multiplication of Rational Expressions

Definition

Multiplication of rational expressions involves finding the product of two or more rational expressions, which are fractions with polynomial numerators and denominators. This operation is crucial in simplifying and manipulating algebraic expressions involving ratios of polynomials.

5 Must Know Facts For Your Next Test

1. The product of two rational expressions is obtained by multiplying the numerators and multiplying the denominators.
2. When multiplying rational expressions, the degree of the resulting numerator is the sum of the degrees of the individual numerators, and the degree of the resulting denominator is the sum of the degrees of the individual denominators.
3. Rational expressions can be multiplied even if the variables in the numerator and denominator are different, as long as the expressions are in the correct form.
4. Multiplying a rational expression by a constant is equivalent to multiplying the numerator by that constant and leaving the denominator unchanged.
5. The properties of exponents apply when multiplying rational expressions, allowing for the simplification of the resulting expression.

Review Questions

• Explain the step-by-step process for multiplying two rational expressions.
  • To multiply two rational expressions, first multiply the numerators together, then multiply the denominators together. The resulting expression will have the numerator as the product of the original numerators and the denominator as the product of the original denominators. Finally, simplify the resulting expression by factoring the numerator and denominator and canceling any common factors.
• Describe how the degree of the resulting numerator and denominator are affected when multiplying rational expressions.
  • When multiplying rational expressions, the degree of the resulting numerator is the sum of the degrees of the individual numerators, and the degree of the resulting denominator is the sum of the degrees of the individual denominators. This is because the multiplication of polynomials follows the rules of exponents, where the exponent of the product is the sum of the exponents of the individual factors.
• Analyze the relationship between multiplying a rational expression by a constant and the resulting expression.
  • Multiplying a rational expression by a constant is equivalent to multiplying the numerator by that constant and leaving the denominator unchanged. This is because the constant can be factored out of the numerator, and the denominator remains unaffected. This property allows for the simplification of rational expressions by factoring out common constants from the numerator and denominator.

Related terms

Rational Expression:

A rational expression is a quotient of two polynomials, where the denominator is not equal to zero.

Polynomial:

A polynomial is an expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers.

Simplifying Rational Expressions: Simplifying a rational expression involves reducing the expression to its lowest terms by factoring the numerator and denominator and canceling common factors.