A rational expression is a fraction where both the numerator and the denominator are polynomials. An example is x3+5xx2−7.
Sometimes the numerator and denominator can be named using function notation. For example, suppose p(x)=x2−7 and q(x)=x3+5x. Rational expressions are undefined for values of x that make the denominator equal 0. Therefore, q(x) cannot equal 0.Write the rational expression in its simplest form. 3x(x+5)3x2−75
Since rational expressions are essentially fractions, it's possible to add, subtract, multiply and divide them. When creating a multiple of a rational expression, q(x)p(x), by multiplying by the factor k in both the denominator and the numerator, the equality still holds true.
q(x)p(x)=q(x)⋅kp(x)⋅k
Canceling out a factor k, yields the same equality. In both cases, the factor k can take all values except 0.
q(x)p(x)=q(x)/kp(x)/k
It is also possible to create a multiple or cancel out a factor by using a more complex polynomial: x(x−1)(x−1)=x1. Consider the domain. The first expression is undefined for x=1, but the second expression is not. It looks like the domain has been expanded when (x−1) was canceled out, but this is not the case. For the equality to hold true, all x-values must give the same value on both sides. Taking this into account gives
x(x−1)x−1=x1,x=1,0.Multiplying rational expressions works the same as multiplying fractions. The numerators and denominators are multiplied separately.
q(x)p(x)⋅g(x)h(x)=q(x)⋅g(x)p(x)⋅h(x)
To divide two rational expressions, the first step is to invert the expression in the denominator, and then multiply, similar to dividing fractions.
q(x)p(x)/g(x)h(x)=q(x)p(x)⋅h(x)g(x)
When rewriting a division of a rational expression as multiplication, it might appear to change the domain. For example, r(x)=x−3x+1/x+7x−9 is undefined for the x-values 3, -7 and 9. Each of these x-values result in the expression in any of the denominators being equal to 0. The rewritten expression q(x)=(x−3)(x−9)(x+1)(x+7)
is, however, undefined only for the x-values 3 and 9. In order to be able to have an equality between the expressions they must have the same domain. Therefore, r(x) and q(x) are equal for all x except -7.Determine the product and quotient of the rational expressions. Simplify completely. x+2x2−1andx+12x
When adding and subtracting rational expressions, the same rules apply as when adding and subtracting fractions. If they share a denominator, the numerators can be added or subtracted directly.
q(x)p(x)+q(x)h(x)=q(x)p(x)+h(x)
q(x)p(x)−q(x)h(x)=q(x)p(x)−h(x)
Determine the sum of the rational expressions. xx+1andx−12x