Rational Expressions MultiplyingDividing AddingSubtracting Complex Fractions Multiplying Dividing

Rational Expressions • Multiplying/Dividing • Adding/Subtracting • Complex Fractions

Multiplying / Dividing Let’s first review how we multiply and divide ordinary fractions. l Do we need a common denominator? l No!

Multiplying / Dividing l How do we multiply ordinary fractions? • Multiply across: numerators times numerators and denominators times denominators.

Multiplying / Dividing l How do we multiply rational expressions with variables? • Multiply across: numerators times numerators and denominators times denominators.

Multiplying / Dividing l For Example: Multiply Across Using Power Rules!

Multiplying / Dividing l l When do we cancel things in fractions? We can only cancel identical factors that appear in both the numerator and denominator. • (x - 3) can only be cancelled by (x - 3), not by x, not by 3.

Multiplying / Dividing 3 Here, 3, 9, (x+5) and (x + 5) are all identical factors and can be cancelled. (2 x-7) and (7 x-2) are factors, but they aren’t identical; we can’t cancel any part of them!

Multiplying / Dividing l l To simplify: We must factor first, then we cancel:

Multiplying / Dividing l Simplify: factor, then cancel

Multiplying / Dividing l When multiplying rational expressions • factor each numerator and denominator first • then cancel identical factors • then multiply across: numerators by numerators and denominators by denominators.

Multiplying / Dividing l Practice:

Multiplying / Dividing l Solution: = = =

Multiplying / Dividing Now on to dividing. l This is exactly like multiplying, except for ONE step. l We multiply by the reciprocal of the 2 nd fraction! l

Multiplying / Dividing l l Divide: Change it to multiplication and flip the 2 nd fraction:

Multiplying / Dividing l Divide: Now proceed like a multiplication problem. Factor first, cancel, multiply. = =

Adding/Subtracting l What do we have to do to add or subtract ordinary fractions? • Change one or both fractions so they have the same common denominator.

Adding/Subtracting l l Find the LCD for two fractions with monomial denominators: The key is that the LCD be something we can reach by multiplying each denominator by missing terms.

Adding/Subtracting l l If we multiply the 1 st denominator by 3 a we get: If we multiply the 2 nd denominator by 5 b we get: Same den. (LCD)

Adding/Subtracting l l Once we have the same denominator, we add the numerators: After adding the numerators, try to factor and cancel in the final fraction if possible.

Adding/Subtracting l l Find the LCD for two fractions with polynomial denominators: First we must factor the denominators. . .

Adding/Subtracting l The LCD will need to include at least : • One (x+2) factor from the 1 st fraction • One (x+3) factor from the 1 st fraction • Two (x+2) factors from the 2 nd fraction We don’t need three (x+2) terms, two will satisfy the needs of BOTH fractions!

Adding/Subtracting l Get the LCD: (x+2)(x+3) (x+2) (x+3)

Adding/Subtracting l Subtract the numerators: = l We cannot factor the numerator, so we are finished (don’t try to cancel anything).

Adding/Subtracting l Practice:

Adding/Subtracting l Practice: Factor: LCD must contain at least: a multiple of 2, a multiple of 4, a factor of (x-3). 2* 2*

Adding/Subtracting Subtract: Here, we do have factors to cancel:

Complex Fractions l Complex fractions are those fractions whose numerators &/or denominators contain fractions. • To simplify them, we just multiply the top & bottom by the LCD.

Complex Fractions l Example What would the LCD be? The denominators are 3 and 6, the LCD is 6.

Complex Fractions l Multiply the top & bottom both by 6: 2 *6 *6

Complex Fractions l Simplify:

Complex Fractions l What would the LCD be? The denominators are y and x, the LCD is xy.

Complex Fractions l Multiply top & bottom by LCD:

Complex Fractions l l The final answer is: We cannot cancel any terms in this fraction!