11.4 – Multiplying & Dividing Rational Functions.

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Presentation transcript:

11.4 – Multiplying & Dividing Rational Functions

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 2

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 2

Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 4wy 5xz 2 2

b. x 2 – x – 20. x x + 22 x x + 99 x 2 + 6x + 8

b. x 2 – x – 20. x x + 22 x x + 99 x 2 + 6x + 8 (x – 5)(x + 4)

b. x 2 – x – 20. x x + 22 x x + 99 x 2 + 6x + 8 (x – 5)(x + 4) (x + 9)(x + 11)

b. x 2 – x – 20. x x + 22 x x + 99 x 2 + 6x + 8 (x – 5)(x + 4). (x + 11)(x + 2) (x + 9)(x + 11)

b. x 2 – x – 20. x x + 22 x x + 99 x 2 + 6x + 8 (x – 5)(x + 4). (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)

b. x 2 – x – 20. x x + 22 x x + 99 x 2 + 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. x 2 – x – 20. x x + 22 x x + 99 x 2 + 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. x 2 – x – 20. x x + 22 x x + 99 x 2 + 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. x 2 – x – 20. x x + 22 x x + 99 x 2 + 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 – 5 x + 9.

Ex. 2 Find the product centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard

Ex. 2 Find the product centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet)

Ex. 2 Find the product 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 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 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 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 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. 16n 4 ÷ 12n 2 9p 2 3p 3

Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3

Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2

Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n

Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p

Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p

Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 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. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 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. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 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. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n 4n 2 p 9

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3)

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3) (x + 3)(x – 3)

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2)

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) (x + 4)(x – 3)(x + 3)

b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 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. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 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. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 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. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 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. ( x 2 + 6x – 16) ÷ (x – 2)

x 2 + 6x – 16

c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2

c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2 (x – 2)(x + 8) x – 2

c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2 (x – 2)(x + 8) x – 2

c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2 (x – 2)(x + 8) x – 2 x + 8