Rational Expressions

Domain — what values of x make the statement valid?  Usually the domain is all real numbers but …. or all real #’s such that x does not equal ….

Examples:                                            Domain:

a)\, \, 2x^{3}+3x+4                                       all reals

b)\, \, \sqrt{x-2}                                               all reals such that   x\geq 2  [cannot be a negative                                                                                                            #  under a \sqrt{\, \, \, \, \, }  sign]

c)\, \, \frac{x+2}{x-3}                                                  all reals such that  x\neq 3  [cannot be 0 in the                                                                                                                denominator]

Simplifying Rational Expressions:

*Be sure to factor each polynomial completely before cancelling common factors.

Ex.           \frac{12+x-x^{2}}{2x^{2}-9x+4}=\frac{\left ( 4-x \right )\left ( 3+x \right )}{\left ( 2x-1 \right )\left ( x-4 \right )}=\frac{-1\left ( x+3 \right )}{2x-1}

Note:     \frac{\left ( x-4 \right )}{\left ( x-4 \right )}=1\; and \; \frac{(4-x)}{(x-4)}=-1

Operations with Rational Expressions

Multiplying – Factor then cancel

Dividing – Change to multiplying / Factor / Cancel

Adding / Subtracting – find the LCD -   \frac{a}{b}\pm \frac{c}{d}=\frac{ad\pm bc }{bd}

Examples:    a)\; \frac{x^{3}+5x^{2}+6x}{x^{2}-4}=\frac{x\left ( x^{2}+5x+6 \right )}{\left ( x+2 \right )\left ( x-2 \right )}=\frac{x\left ( x+3 \right )\left ( x+2 \right )}{\left ( x+2 \right )\left ( x-2 \right )}= 

                           \frac{x\left ( x+3 \right )}{x-2},\; x\neq 2,-2

                      b)\; \frac{x^{2}+xy-2y^{2}}{x^{3}+x^{2}y}*\frac{x}{x^{2}+3xy+2y^{2}}

                       =\frac{\left ( x+2y \right )\left ( x-y \right )}{x\cdot x\left ( x+y \right )}*\frac{x}{\left ( x+2y \right )\left ( x+y \right )}

                       =\frac{x-y}{x\left ( x+y \right )^{2}},\; x\neq 0,-y,-2y

                    c)\; \frac{x^{2}-36}{x}\div \frac{x^{3}-6x^{2}}{x^{2}+x}

                       =\frac{\left ( x+6 \right )\left ( x-6 \right )}{x}*\frac{x\left ( x+1 \right )}{x\cdot x\left ( x-6 \right )}

                      =\frac{\left ( x+6 \right )\left ( x+1 \right )}{x^{2}},\; x\neq 0,6

                    d)\; \frac{2}{x^{2}-4}-\frac{1}{x^{2}-3x+2}=\frac{2}{\left ( x+2 \right )\left ( x-2 \right )}-\frac{1}{\left ( x-2 \right )\left ( x-1 \right )}

                       =\frac{2\left ( x-1 \right )-1\left ( x+2 \right )}{\left ( x+2 \right )\left ( x-2 \right )\left ( x-1 \right )}=\frac{2x-2-x-2}{\left ( x+2 \right )\left ( x-2 \right )\left ( x-1 \right )}

                       =\frac{x-4}{\left ( x+2 \right )\left ( x-2 \right )\left ( x-1 \right )},\; x\neq \pm 2,1

Complex Fractions —

Fractions in the numerator and/or in the denominator

Examples:    a)\; \frac{\frac{x}{2}-1}{x-2}=\left ( \frac{x}{2}-\frac{2}{2} \right )\div \left ( x-2 \right )=\frac{x-2}{2}*\frac{1}{x-2}=\frac{1}{2},\; x\neq 2

                      b)\; \left ( \sqrt{x} \right -\frac{1}{2\sqrt{x}})\div \sqrt{x}= \left ( \frac{2x-1}{2\sqrt{x}} \right )*\frac{1}{\sqrt{x}}=\frac{2x-1}{2x},\; x\nleqslant 0

Filed under: Pre-requisites for Pre-Calculus

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