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Review: Rewriting and Simplifying Fractions Simplifying Rational Expressions Simplify: 2x 3x2 0 4x36 4x 2 2x5x4 4xx216 This form is more convenient in order to find the domain 2x5x4 4 xx 4x4 Can NOT cancel since everything does not have a common factor and its not in factored form Factor Completely CAN cancel since the top and bottom have a common factor 2x5 2 x5 o r 2 4xx4 4 x 1 6 x Polynomial Division: Area Method Simplify: 4 2 x 1 0 x 2 x 3 x 3 Divisor Quotient x x3 3x2 -x -1 x4 3x3 -x2 -x The sum of these boxes must be the dividend -3 x 4 Needed 3 -3x 2 -9x 3x 3 –10x +2x +3 Check Needed 3 Needed2 Needed x + 3x – x – 1 3 +0x 2 Dividend (make sure to include all powers of x) Rationalizing Irrational and Complex Denominators The denominator of a fraction typically can not contain an imaginary number or any other radical. To rationalize the denominator (rewriting a fraction so the bottom is a rational number) multiply by the conjugate of the denominator. Ex: Rationalize the denominator of each fraction. a. 4 6 7 6 7 6 7 x b. x7 7 2 4 47 3 6 67 67 7 x 7 7 x 7 7 244 7 29 xx 7 7 x 4 7x 7 7x 7 4 x7 7 x x 7 7 x Simplifying Complex Fractions xy 1 x y 1 xy x Simplify: xy 1 x y 1 xy y xy 1 x y x It is not simplified since xy it has embedded 1 xy y fractions xy y xy x 1 x 1 y Check to see if it can be simplified more: xy y y x 1 xy x x y 1 To eliminate the denominators of the embedded fractions, multiply by a common denominator No Common Factor. Not everything can be simplified! Trigonometric Identities tan x cot x tan x cot x Simplify: sin x cos x sin x cos x sin x cos x 1 1 tan x cot x sin x cos x sin x cos x sin x 1 cos x 1 cos x sin x cos x sin x sin x cos x 1 1 2 2 cos x sin x Split the fraction Use Trigonometric Identities sec x csc x 2 2 Write as simple as possible