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Conjugate Zero Theorem

Conjugate Zero Theorem. OBJ:  To find the complex zeros of an integral polynomial using the conjugate zero theorem. DEF:  Conjugate Zero Theorem. If a + bi is a zero of an integral polynomial P(x) then a – bi is also a zero.

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Conjugate Zero Theorem

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  1. Conjugate Zero Theorem OBJ:  To find the complex zeros of an integral polynomial using the conjugate zero theorem

  2. DEF: Conjugate Zero Theorem If a + bi is a zero of an integral polynomial P(x) then a – bi is also a zero.

  3. P 419 HW 5 P 419 (10 – 13) EX 1: P(x) = x4–5x3+15x2–5x– 26 2 –3 i is a zero of P( x ) 2 – 3 i| 1 -5 +15 5 -26 2 – 3 i 1 -3 – 3 i (2 – 3i)(-3 – 3i) -6 – 6i + 9i + 9i2 -15 + 3i 2 – 3 i| 1 -5 +15 -5 -26 2 – 3 i -15 + 3i 1 -3 – 3 i 3i (2 – 3i)(3i) 6i – 9i2 2 – 3 i| 1 -5 +15 -5 -26 2 – 3 i -15 + 3i 9 + 6i 1 -3 – 3 i 3i 4 + 6i (2 – 3i)(4 + 6i) 8 + 12i – 12i – 18i2 26 2 – 3 i| 1 -5 +15 -5 -26 2 – 3 i -15 + 3i 9 + 6i 26 1 -3 – 3 i 3i 4 + 6i 0

  4. P 419 HW 5 P 419 (10 – 13) EX 1: P(x) = x4–5x3+15x2–5x– 26 2 + 3 i| 1 -3 – 3 i 3i 4 + 6i 2 + 3 i 1 -1 2 + 3 i| 1 -3 – 3 i 3i 4 + 6i 2 + 3 i -2 – 3i 1 -1 -2 2 + 3 i| 1 -3 – 3 i 3i 4 + 6i 2 + 3 i -2 – 3i -4 – 6i 1 -1 -2 | 0 | x2 – x – 2 = 0 (x – 2)(x + 1) = 0 x = 2, -1

  5. P415 214x4 –4x3 +17x2 –16x +4 ± 1, 2, 4, 1/2, 1/4 1/2| 4 -4 17 -16 4 2 -1 8 -4 4 -2 16 -8 0 4x3 – 2x2 + 16x – 8 = 0 2x2(2x – 1) + 8(2x – 1) = 0 (2x – 1)(2x2 + 8) = 0 x = 1/2 2x2 + 8 = 0 2x2 = -8 x2 = -4 x = ±2i

  6. P415 2336x4 –12x3 -11x2 +2x + 1 ± 1, ½, 1/3, ¼, 1/6, 1/9, 1/12, 1/18, 1/36 1/2| 36 -12 -11 +2 +1 18 3 -4 -1 -1/336 6 -8 -2 0 -12 2 2 36 -6 -6 0 36x2 – 6x – 6) = 0 6(6x2 – x – 1) = 0 6(3x + 1 )(2x – 1)=0 6(3x + 1 )(2x – 1)(x – ½)(x + 1/3) (3x + 1 )(2x – 1) (3x + 1 )(2x – 1)

  7. P415 25 6x5 + x4 +20x3 + 5x2 – 16x + 4 ±1, 2, 4, ½, 1/3, 2/3, 4/3, 1/6 -1| 6 1 20 5-16 4 -6 5 -25 20 -4 ½ |6 -5 25 -20 4 0 3 -1 12 -4 1/3|6 -2 24 -8 0 2 0 8 6 0 24 0 6x2 + 24 = 0 6x2 = -24 x2 = -4 x = ±2i

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