11 x1 t15 01 polynomial definitions (2012)

Polynomial Functions

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where np

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

coefficients: 0 1 2, , , , np p p p

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

coefficients: 0 1 2, , , , np p p p

index (exponent): the powers of the pronumerals.

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n”

coefficients: 0 1 2, , , , np p p p

index (exponent): the powers of the pronumerals.

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n”

coefficients: 0 1 2, , , , np p p p

index (exponent): the powers of the pronumerals.

leading term: nnp x

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n”

leading coefficient:

coefficients: 0 1 2, , , , np p p p

index (exponent): the powers of the pronumerals.

leading term: nnp x

np

Polynomial FunctionsA real polynomial P(x) of degree n is an expression of the form;

nn

nn xpxpxpxppxP

11

2210

0 :where npintegeran isand 0n

degree (order): the highest index of the polynomial. The polynomial is called “polynomial of degree n”

leading coefficient: monic polynomial: leading coefficient is equal to one.

coefficients: 0 1 2, , , , np p p p

index (exponent): the powers of the pronumerals.

leading term: nnp x

np

P(x) = 0: polynomial equation

P(x) = 0: polynomial equationy = P(x): polynomial function

roots: solutions to the polynomial equation P(x) = 0

P(x) = 0: polynomial equationy = P(x): polynomial function

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

YES, 07x

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

YES, 07x(ii) Determine whether is

monic and state its degree. 3 2 2( ) 8 1 7 11 2 1 4 3P x x x x x x

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

YES, 07x(ii) Determine whether is

monic and state its degree. 3 2 2( ) 8 1 7 11 2 1 4 3P x x x x x x

4 3 4 2 2( ) 8 7 11 8 6 4 3P x x x x x x x

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

YES, 07x(ii) Determine whether is

monic and state its degree. 3 2 2( ) 8 1 7 11 2 1 4 3P x x x x x x

4 3 4 2 2( ) 8 7 11 8 6 4 3P x x x x x x x 3 22 7 8x x x

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

YES, 07x(ii) Determine whether is

monic and state its degree. 3 2 2( ) 8 1 7 11 2 1 4 3P x x x x x x

4 3 4 2 2( ) 8 7 11 8 6 4 3P x x x x x x x 3 22 7 8x x x monic, degree = 3

roots: solutions to the polynomial equation P(x) = 0zeros: the values of x that make polynomial P(x) zero. i.e. the x intercepts of the graph of the polynomial.

P(x) = 0: polynomial equationy = P(x): polynomial function

e.g. (i) Which of the following are polynomials?1

3 2

2

2

a) 5 7 24b)

33c)

4d) 7

x x

xx

NO, can’t have fraction as a power

NO, can’t have negative as a power 124 3x

YES,21 3

4 4x

YES, 07x(ii) Determine whether is

monic and state its degree. 3 2 2( ) 8 1 7 11 2 1 4 3P x x x x x x

4 3 4 2 2( ) 8 7 11 8 6 4 3P x x x x x x x 3 22 7 8x x x monic, degree = 3

Exercise 4A; 1, 2acehi, 3bdf, 6bdf, 7, 9d, 10ad, 13