The document defines and explains key concepts related to polynomial functions. A polynomial P(x) of degree n is an expression of the form P(x) = p0 + p1x + p2x2 + ... + pn-1xn-1 + pnxn, where pn ≠ 0. The degree of a polynomial is the highest exponent in the polynomial. Other important terms defined include coefficients, leading term, leading coefficient, monic polynomials, roots, and zeros. Examples are provided to demonstrate identifying polynomials and determining the degree and form of a given polynomial.

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