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