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11X1 T16 01 definitions

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  • 1. Polynomial Functions
  • 2. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n
  • 3. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0
  • 4. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer
  • 5. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer coefficients: p0 , p1 , p2 , , pn
  • 6. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer coefficients: p0 , p1 , p2 , , pn index (exponent): the powers of the pronumerals.
  • 7. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer coefficients: p0 , p1 , p2 , , pn index (exponent): the powers of the pronumerals. degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n”
  • 8. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer coefficients: p0 , p1 , p2 , , pn index (exponent): the powers of the pronumerals. degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n” n leading term: pn x
  • 9. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer coefficients: p0 , p1 , p2 , , pn index (exponent): the powers of the pronumerals. degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n” n leading term: pn x leading coefficient: pn
  • 10. Polynomial Functions A real polynomial P(x) of degree n is an expression of the form; P x   p0  p1 x  p2 x 2    pn1 x n1  pn x n where : pn  0 n  0 and is an integer coefficients: p0 , p1 , p2 , , pn index (exponent): the powers of the pronumerals. degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n” n leading term: pn x leading coefficient: pn monic polynomial: leading coefficient is equal to one.
  • 11. P(x) = 0: polynomial equation
  • 12. P(x) = 0: polynomial equation y = P(x): polynomial function
  • 13. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0
  • 14. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.
  • 15. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 4 b) 2 x 3 x2  3 c) 4 d) 7
  • 16. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 b) 2 x 3 x2  3 c) 4 d) 7
  • 17. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 c) 4 d) 7
  • 18. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  4 4 4 d) 7
  • 19. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  4 4 4 d) 7 YES, 7x 0
  • 20. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  4 4 4 d) 7 YES, 7x 0 (ii) Determine whether P( x)  x 3  8 x  1  7 x  11   2 x 2  1 4 x 2  3 is monic and state its degree.
  • 21. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  4 4 4 d) 7 YES, 7x 0 (ii) Determine whether P( x)  x 3  8 x  1  7 x  11   2 x 2  1 4 x 2  3 is monic and state its degree. P( x)  8 x 4  x3  7 x  11  8 x 4  6 x 2  4 x 2  3
  • 22. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  4 4 4 d) 7 YES, 7x 0 (ii) Determine whether P( x)  x 3  8 x  1  7 x  11   2 x 2  1 4 x 2  3 is monic and state its degree. P( x)  8 x 4  x3  7 x  11  8 x 4  6 x 2  4 x 2  3  x3  2 x 2  7 x  8
  • 23. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  4 4 4 d) 7 YES, 7x 0 (ii) Determine whether P( x)  x 3  8 x  1  7 x  11   2 x 2  1 4 x 2  3 is monic and state its degree. P( x)  8 x 4  x3  7 x  11  8 x 4  6 x 2  4 x 2  3  x3  2 x 2  7 x  8  monic, degree = 3
  • 24. P(x) = 0: polynomial equation y = P(x): polynomial function roots: solutions to the polynomial equation P(x) = 0 zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial. e.g. (i) Which of the following are polynomials? 1 a) 5 x 3  7 x  2 2 NO, can’t have fraction as a power 4 NO, can’t have negative as a power 4  x  3 1 b) 2 2 x 3 x2  3 1 2 3 c) YES, x  Exercise 4A; 1, 2acehi, 3bdf, 4 4 4 6bdf, 7, 9d, 10ad, 13 0 d) 7 YES, 7x (ii) Determine whether P( x)  x 3  8 x  1  7 x  11   2 x 2  1 4 x 2  3 is monic and state its degree. P( x)  8 x 4  x3  7 x  11  8 x 4  6 x 2  4 x 2  3  x3  2 x 2  7 x  8  monic, degree = 3