# 11x1 t16 05 polynomial results

Post on 01-Dec-2014

1.003 views

Embed Size (px)

DESCRIPTION

TRANSCRIPT

- 1. Polynomial Results
- 2. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x).
- 3. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x).
- 4. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0
- 5. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0
- 6. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0 1, 2 are zeros of P ( x) and x 1 x 2 is a factor
- 7. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0 1, 2 are zeros of P ( x) and x 1 x 2 is a factor P( x) x 4 x 3 3 x 2 5 x 10
- 8. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0 1, 2 are zeros of P ( x) and x 1 x 2 is a factor P( x) x 4 x 3 3 x 2 5 x 10 x2 x 2
- 9. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0 1, 2 are zeros of P ( x) and x 1 x 2 is a factor P( x) x 4 x 3 3 x 2 5 x 10 x2 x 2 x2
- 10. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0 1, 2 are zeros of P ( x) and x 1 x 2 is a factor P( x) x 4 x 3 3 x 2 5 x 10 x2 x 2 x2 5
- 11. Polynomial Results 1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then; x a1 x a2 x a3 x ak is a factor of P(x). e.g. Show that 1 and 2 are zeros of P( x) x 4 x 3 3 x 2 5 x 10 and hence factorise P(x). P 1 1 1 3 1 5 1 10 4 3 2 0 P 2 2 2 3 2 5 2 10 4 3 2 0 1, 2 are zeros of P ( x) and x 1 x 2 is a factor P( x) x 4 x 3 3 x 2 5 x 10 x2 x 2 x2 5 x 1 x 2 x 2 5
- 12. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an
- 13. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros
- 14. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros e.g. The polynomial P(x) has a double zero at 7 and a single zero at 2. Write down; a) a possible polynomial
- 15. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros e.g. The polynomial P(x) has a double zero at 7 and a single zero at 2. Write down; a) a possible polynomial P( x) x 2 x 7 2
- 16. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros e.g. The polynomial P(x) has a double zero at 7 and a single zero at 2. Write down; a) a possible polynomial P( x) x 2 x 7 Q( x) 2 where - Q(x) is a polynomial and does not have a zero at 2 or 7
- 17. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros e.g. The polynomial P(x) has a double zero at 7 and a single zero at 2. Write down; a) a possible polynomial P( x) k x 2 x 7 Q( x) 2 where - Q(x) is a polynomial and does not have a zero at 2 or 7 - k is a real number
- 18. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros e.g. The polynomial P(x) has a double zero at 7 and a single zero at 2. Write down; a) a possible polynomial P( x) k x 2 x 7 Q( x) 2 where - Q(x) is a polynomial and does not have a zero at 2 or 7 - k is a real number b) a monic polynomial of degree 3.
- 19. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an , then P x x a1 x a2 x a3 x an 3. A polynomial of degree n cannot have more than n distinct real zeros e.g. The polynomial P(x) has a double zero at 7