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11.4 – Multiplying & Dividing Rational Functions

11.4 – Multiplying & Dividing Rational Functions

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Page 1: 11.4 – Multiplying & Dividing Rational Functions

11.4 – Multiplying & Dividing Rational Functions

Page 2: 11.4 – Multiplying & Dividing Rational Functions

Ex. 1 Find each product.a. 10y3z2 . 12w2x2

6wx3 25y2z4

Page 3: 11.4 – Multiplying & Dividing Rational Functions

Ex. 1 Find each product.a. 10y3z2 . 12w2x2

6wx3 25y2z4

10y3z2 · 12w2x2

Page 4: 11.4 – Multiplying & Dividing Rational Functions

Ex. 1 Find each product.a. 10y3z2 . 12w2x2

6wx3 25y2z4

10y3z2 · 12w2x2

6wx3 · 25y2z4

Page 5: 11.4 – Multiplying & Dividing Rational Functions

Ex. 1 Find each product.a. 10y3z2 . 12w2x2

6wx3 25y2z4

10y3z2 · 12w2x2

6wx3 · 25y2z4

2 · 5 · y · y · y · z · z

Page 6: 11.4 – Multiplying & Dividing Rational Functions

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

Page 7: 11.4 – Multiplying & Dividing Rational Functions

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

Page 8: 11.4 – Multiplying & Dividing Rational Functions

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

Page 9: 11.4 – Multiplying & Dividing Rational Functions

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

Page 10: 11.4 – Multiplying & Dividing Rational Functions

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

Page 11: 11.4 – Multiplying & Dividing Rational Functions

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

Page 12: 11.4 – Multiplying & Dividing Rational Functions

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

Page 13: 11.4 – Multiplying & Dividing Rational Functions

b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8

Page 14: 11.4 – Multiplying & Dividing Rational Functions

b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8

(x – 5)(x + 4)

Page 15: 11.4 – Multiplying & Dividing Rational Functions

b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8

(x – 5)(x + 4) (x + 9)(x + 11)

Page 16: 11.4 – Multiplying & Dividing Rational Functions

b. x2 – x – 20 . x2 + 13x + 22x2 + 20x + 99 x2 + 6x + 8

(x – 5)(x + 4) . (x + 11)(x + 2) (x + 9)(x + 11)

Page 17: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 18: 11.4 – Multiplying & Dividing Rational Functions

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)

..

Page 19: 11.4 – Multiplying & Dividing Rational Functions

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)

..

Page 20: 11.4 – Multiplying & Dividing Rational Functions

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)

..

Page 21: 11.4 – Multiplying & Dividing Rational Functions

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

..

Page 22: 11.4 – Multiplying & Dividing Rational Functions

Ex. 2 Find the product.2.54 centimeters . 12 inches . 3 feet

1 inch 1 foot 1 yard

Page 23: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 24: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 25: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 26: 11.4 – Multiplying & Dividing Rational Functions

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.

Page 27: 11.4 – Multiplying & Dividing Rational Functions

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.

Page 28: 11.4 – Multiplying & Dividing Rational Functions

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.

Page 29: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?

Page 30: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile

Page 31: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet 1 mile

Page 32: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet 1 mile

Page 33: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · = inches

1 mile

Page 34: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · 12 inches

1 mile 1 foot

Page 35: 11.4 – Multiplying & Dividing Rational Functions

Ex. 3 How many inches are in a mile?Note: 5280 feet = 1 mile 5280 feet · 12 inches

1 mile 1 foot

Page 36: 11.4 – Multiplying & Dividing Rational Functions

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

Page 37: 11.4 – Multiplying & Dividing Rational Functions

Ex. Find each quotient.a. 16n4 ÷ 12n2

9p2 3p3

Page 38: 11.4 – Multiplying & Dividing Rational Functions

Ex. Find each quotient.a. 16n4 ÷ 12n2

9p2 3p3

Page 39: 11.4 – Multiplying & Dividing Rational Functions

Ex. Find each quotient.a. 16n4 ÷ 12n2

9p2 3p3

16n4 . 3p3

9p2 12n2

Page 40: 11.4 – Multiplying & Dividing Rational Functions

Ex. Find each quotient.a. 16n4 ÷ 12n2

9p2 3p3

16n4 . 3p3

9p2 12n2

2 · 2 · 2 · 2 · n · n · n · n

Page 41: 11.4 – Multiplying & Dividing Rational Functions

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

Page 42: 11.4 – Multiplying & Dividing Rational Functions

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

Page 43: 11.4 – Multiplying & Dividing Rational Functions

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

Page 44: 11.4 – Multiplying & Dividing Rational Functions

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

Page 45: 11.4 – Multiplying & Dividing Rational Functions

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

Page 46: 11.4 – Multiplying & Dividing Rational Functions

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

Page 47: 11.4 – Multiplying & Dividing Rational Functions

b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3

Page 48: 11.4 – Multiplying & Dividing Rational Functions

b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3

Page 49: 11.4 – Multiplying & Dividing Rational Functions

b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2

Page 50: 11.4 – Multiplying & Dividing Rational Functions

b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2

Page 51: 11.4 – Multiplying & Dividing Rational Functions

b. x2 + x – 12 ÷ x + 2 x2 – 9 x + 3 x2 + x – 12 . x + 3 x2 – 9 x + 2

(x + 4)(x – 3)

Page 52: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 53: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 54: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 55: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 56: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 57: 11.4 – Multiplying & Dividing Rational Functions

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)

Page 58: 11.4 – Multiplying & Dividing Rational Functions

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

Page 59: 11.4 – Multiplying & Dividing Rational Functions

c. (x2 + 6x – 16) ÷ (x – 2)

Page 60: 11.4 – Multiplying & Dividing Rational Functions

c. (x2 + 6x – 16) ÷ (x – 2)

x2 + 6x – 16

Page 61: 11.4 – Multiplying & Dividing Rational Functions

c. (x2 + 6x – 16) ÷ (x – 2)

x2 + 6x – 16

x – 2

Page 62: 11.4 – Multiplying & Dividing Rational Functions

c. (x2 + 6x – 16) ÷ (x – 2)

x2 + 6x – 16

x – 2

(x – 2)(x + 8)

x – 2

Page 63: 11.4 – Multiplying & Dividing Rational Functions

c. (x2 + 6x – 16) ÷ (x – 2)

x2 + 6x – 16

x – 2

(x – 2)(x + 8)

x – 2

Page 64: 11.4 – Multiplying & Dividing Rational Functions

c. (x2 + 6x – 16) ÷ (x – 2)

x2 + 6x – 16

x – 2

(x – 2)(x + 8)

x – 2

x + 8