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# 11 x1 t15 05 polynomial results (2013)

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### 11 x1 t15 05 polynomial results (2013)

1. 1. Polynomial Results
2. 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. 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. 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 0 3 2
5. 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 4 3 2 P  2    2    2   3  2   5  2   10 0
6. 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 4 3 2 P  2    2    2   3  2   5  2   10 0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor
7. 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 4 3 2 P  2    2    2   3  2   5  2   10 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. 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 4 3 2 P  2    2    2   3  2   5  2   10 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. 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 4 3 2 P  2    2    2   3  2   5  2   10 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. 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 4 3 2 P  2    2    2   3  2   5  2   10 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. 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 4 3 2 P  2    2    2   3  2   5  2   10 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   x  1 x  2   x 2  5  5 
12. 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. 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. 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. 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. 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) where - Q(x) is a polynomial and does not have a zero at 2 or –7 2
17. 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) where - Q(x) is a polynomial and does not have a zero at 2 or –7 - k is a real number 2
18. 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) where - Q(x) is a polynomial and does not have a zero at 2 or –7 - k is a real number 2 b) a monic polynomial of degree 3.
19. 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 and a single zero at 2. Write down; a) a possible polynomial P( x)  k  x  2  x  7  Q( x) where - Q(x) is a polynomial and does not have a zero at 2 or –7 - k is a real number 2 b) a monic polynomial of degree 3. P( x)   x  2  x  7  2
20. 20. c) A monic polynomial of degree 4
21. 21. c) A monic polynomial of degree 4 P( x)   x  2  x  7  2
22. 22. c) A monic polynomial of degree 4 P( x)   x  2  x  7   x  a  where a  2 or  7 d) a polynomial of degree 5. 2
23. 23. c) A monic polynomial of degree 4 P( x)   x  2  x  7   x  a  2 where a  2 or  7 d) a polynomial of degree 5. P( x)   x  2  x  7  2
24. 24. c) A monic polynomial of degree 4 P( x)   x  2  x  7   x  a  2 where a  2 or  7 d) a polynomial of degree 5. P( x)   x  2  x  7  Q( x) 2 where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7
25. 25. c) A monic polynomial of degree 4 P( x)   x  2  x  7   x  a  2 where a  2 or  7 d) a polynomial of degree 5. P( x)  k  x  2  x  7  Q( x) 2 where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7 - k is a real number
26. 26. c) A monic polynomial of degree 4 P( x)   x  2  x  7   x  a  2 where a  2 or  7 d) a polynomial of degree 5. P( x)  k  x  2  x  7  Q( x) 2 where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7 - k is a real number 4. If P(x) has degree n and has more than n real zeros, then P(x) is the zero polynomial. i.e. P(x) = 0 for all values of x
27. 27. Exercise 4E; 1, 3bd, 4a, 5b, 6a, 8, 12ad