Simplify Rational Expressions Quiz

PDF page 912: A rational expression is an expression of the form , where p and q are polynomials and q ≠ 0. q(x)

PDF page 912: In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.

PDF page 912: A rational expression is an expression of the form , where p and q are polynomials and q ≠ 0. q(x)

PDF page 912: Remember, division by 0 is undefined. Here are some examples of rational expressions: 2

PDF page 912: When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero. If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator. So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

PDF page 912: When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero. If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator. So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

PDF page 912: Step 1. Set the denominator equal to zero. Step 2. Solve the equation in the set of reals, if possible.

PDF page 912: HOW TO : : DETERMINE THE VALUES FOR WHICH A RATIONAL EXPRESSION IS UNDEFINED.

PDF page 913: 9y x Set the denominator equal to zero. Solve x = 0 for the variable. 9y is undefined for x = 0. x

PDF page 913: 4b − 3 2b + 5 2b + 5 = 0 Set the denominator equal to zero. Solve 2b = −5 for the variable. 5 b = − 2 4b − 3 5 is undefined for b = − . 2 2b + 5

PDF page 913: x + 5x + 6 = 0 Set the denominator equal to zero. Solve (x + 2)(x + 3) = 0 for the variable.

PDF page 913: is undefined for x = −2 or x = −3.

PDF page 913: To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.

PDF page 917: Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.

PDF page 917: A rational expression is considered simplified if there are no common factors in its numerator and denominator. For example:

PDF page 917: • is not simplified because x is a common factor of 2x and 3x.

PDF page 917: We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.