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.
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