Dividing Polynomials WOW! I want to learn how to do that?

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Presentation transcript:

Dividing Polynomials WOW! I want to learn how to do that? Why would you want to do that? What do you find when you divide 15 by 5? What do you find when you divide ( 6x3 – 5x2 – 12x – 4 ) by (3x+2)? You find 3, the other factor of 15 You find 2x2 – 3x – 2, another factor of the polynomial. Which you can then factor into (2x+1)(x-2)

3x + 2 6x3 - 5x2 – 12x – 4 2x2 – 3x – 2 6x3 + 4x2 -9x2 – 12x -9x2 – 6x D M S B 3x + 2 6x3 - 5x2 – 12x – 4 6x3 + 4x2 -9x2 – 12x -9x2 – 6x – 6x – 4 – 6x – 4 Check: (3x + 2)(2x2 – 3x – 2) = 6x3 – 9x2 – 6x + 4x2 – 6x – 4 6x3 – 5x2 – 12x – 4

2x – 1 4x3 – 8x2 + 5x – 8 2x2 – 3x + 1 4x3 – 2x2 + 5x -6x2 -6x2 + 3x 7 (2x – 1) 2x – 1 4x3 – 8x2 + 5x – 8 4x3 – 2x2 + 5x -6x2 -6x2 + 3x 2x - 8 2x – 1 - 7 remainder Check: (2x – 1)(2x2 – 3x + 1) = 4x3 – 6x2 + 2x – 2x2 + 3x – 1 = 4x3 – 8x2 + 5x – 1 Add remainder -7 = 4x3 – 8x2 + 5x – 8

x2 + 3x 5x4 + 18x3 + 8x2 – 3x 5x2 + 3x – 1 5x4 + 15x3 3x3 + 8x2 3x3

Synthetic Division: Your divisor must be x - # Who’s in the house? ┘ Write the coefficients using zeros if needed. Drop it like it’s hot! (one time) Multiply by the house, then add. (repeat)

(3x3 + 8x2 + 5x – 7)  (x + 2) -2 3 8 5 -7 -6 -4 -2 3 2 1 -9 2 3 1 – Synthetic Division (3x3 + 8x2 + 5x – 7)  (x + 2) -2 3 8 5 -7 -6 -4 -2 3 2 1 -9 remainder 9 x + 2 – answer 2 3 1 x2 + x +

(3x4 + 13x3 + 2x2 - 3x + 20)  (x + 4) -4 3 13 2 -3 20 -12 -4 8 -20 3 Synthetic Division (3x4 + 13x3 + 2x2 - 3x + 20)  (x + 4) -4 3 13 2 -3 20 -12 -4 8 -20 3 1 -2 5 remainder answer 1 3 - 2 5 x3 + x2 x +

2 6 -18 -120 120 12 24 48 60 6 12 24 30 60

Synthetic Division (x4 – 5x2 – 36)  (x – 3) 3 1 -5 -36 3 9 12 36 1 3 -5 -36 3 9 12 36 remainder 1 3 4 12 answer x3 + 3x2 + 4x + 12  (x + 3) -3 1 3 4 12 -3 -12 Imaginary factors 1 4 x2 + 4

Synthetic Division (3x3 – 13x2 – 7x + 2)  (3x + 2) 2 3 -13 3 - 7 3 2 3 1 10 3 - 2 3 -2 3 1 -5 1 remainder answer x2 – 5x + 1

(4x3 – 8x2 + 3x – 8)  (2x – 1) 2 -4 -4 1 2 -3 -4 2x2 – 3x – Synthetic Division (4x3 – 8x2 + 3x – 8)  (2x – 1) 1 2 3 2 2 -4 -4 -3 2 1 2 -3 -4 remainder 4 (2x – 1) answer 2x2 – 3x –

5x4 + 18x3 + 8x2 - 3x 5x2 + 3x – 1 Factor: One factor is x2 + 3x Divide to find that the other factor is: 5x2 + 3x – 1 Connection: the solutions to 5x4 + 18x3 + 8x2 - 3x = 0 are … x = 0 and x = -3 and = x = = .24 and x = = -.84

( 6x2 – x – 7)  (3x + 1) 2 2 -1 -2 2x – 1 – Synthetic Division 1 3 2 -7 3 2 -2 3 1 3 2 -1 -2 remainder answer 2 (3x + 1) 2x – 1 –