11X1 T13 01 polynomial 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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 pn1 x n1 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, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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, cant have fraction as a power 4 NO, cant 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