11.4 – multiplying & dividing rational functions
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11.4 – Multiplying & Dividing Rational Functions
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x3 · 2 · w · x · x · x
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z
2
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z
2
Ex. 1 Find each product.a. 10y3z2 . 12w2x2
6wx3 25y2z4
10y3z2 · 12w2x2
6wx3 · 25y2z4
2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 4wy5xz2
2
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4)
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) (x + 9)(x + 11)
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) . (x + 11)(x + 2) (x + 9)(x + 11)
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) . (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)
(x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4)
..
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)
(x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4)
..
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)
(x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4)
..
b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8
(x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)
(x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4)
x – 5x + 9
..
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
(2.54 centimeters)(12 inches)(3 feet)
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
(2.54 centimeters)(12 inches)(3 feet)(1 inch)(1 foot)(1 yard)
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
(2.54 centimeters)(12 inches)(3 feet)(1 inch)(1 foot)(1 yard)
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
(2.54 centimeters)(12 inches)(3 feet)(1 inch)(1 foot)(1 yard)
(2.54cm.)(12)(3)1 yd.
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
(2.54 centimeters)(12 inches)(3 feet)(1 inch)(1 foot)(1 yard)
(2.54cm.)(12)(3) = 91.44cm.
1 yd. 1 yd.
Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet
1 inch 1 foot 1 yard
(2.54 centimeters)(12 inches)(3 feet)(1 inch)(1 foot)(1 yard)
(2.54cm.)(12)(3) = 91.44cm. = 91.44cm/yd
1 yd. 1 yd.
Ex. 3 How many inches are in a mile?
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet 1 mile
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet 1 mile
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · = inches
1 mile
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · 12 inches
1 mile 1 foot
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · 12 inches
1 mile 1 foot
Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · 12 inches = 63,360 in/mi
1 mile 1 foot
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n
Ex. Find each quotient.a. 16n4 ÷ 12n2
9p2 3p3
16n4 . 3p3
9p2 12n2
2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n
4n2p 9
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) (x + 3)(x – 3)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) . (x + 3) (x + 3)(x – 3) (x + 2)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) . (x + 3) (x + 3)(x – 3) (x + 2)
(x + 4)(x – 3)(x + 3)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) . (x + 3) (x + 3)(x – 3) (x + 2)
(x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) . (x + 3) (x + 3)(x – 3) (x + 2)
(x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) . (x + 3) (x + 3)(x – 3) (x + 2)
(x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)
b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2
(x + 4)(x – 3) . (x + 3) (x + 3)(x – 3) (x + 2)
(x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)
x + 4 x + 2
c. (x2 + 6x – 16) ÷ (x – 2)
c. (x2 + 6x – 16) ÷ (x – 2)
x2 + 6x – 16
c. (x2 + 6x – 16) ÷ (x – 2)
x2 + 6x – 16
x – 2
c. (x2 + 6x – 16) ÷ (x – 2)
x2 + 6x – 16
x – 2
(x – 2)(x + 8)
x – 2
c. (x2 + 6x – 16) ÷ (x – 2)
x2 + 6x – 16
x – 2
(x – 2)(x + 8)
x – 2
c. (x2 + 6x – 16) ÷ (x – 2)
x2 + 6x – 16
x – 2
(x – 2)(x + 8)
x – 2
x + 8
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