Polyn

omials

2x2 + 3x = 5

2x2 + 3x= 9

x 3 – 3x 2 + x +1 = 0

4y3 - 4y2 + 5y + 8 = 0

9x 2 + 9y + 8 =0

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions.

Introduction :

A polynomial is an expression of finite length constructed from

variables and constants, using only the operations of addition,

subtraction, multiplication, and non-negative, whole-number exponents.

Polynomials appear in a wide variety.

Let x be a variable n, be a positive integer and as, a1,a2,….an be constants (real nos.)

Then, f(x) = anxn+ an-1xn-1+….+a1x+xo

anxn,an-1xn-1,….a1x and ao are known as the terms of the polynomial.

an,an-1,an-2,….a1 and ao are their coefficients.

For example:

• p(x) = 3x – 2 is a polynomial in variable x.

• q(x) = 3y2 – 2y + 4 is a polynomial in variable y.

• f(u) = 1/2u3 – 3u2 + 2u – 4 is a polynomial in variable u.

NOTENOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.

The exponent of the highest degree term in a polynomial is known as its degree.

For example:

f(x) = 3x + ½ is a polynomial in the variable x of degree 1.

g(y) = 2y2 – 3/2y + 7 is a polynomial in the variable y of degree 2.

p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in the variable x of degree 3.

q(u) = 9u5 – 2/3u4 + u2 – ½ is a polynomial in the variable u of degree 5.

Polynomials in one variable

A polynomial is a monomial or a sum of monomials.

Each monomial in a polynomial is a term of the polynomial.

The number factor of a term is called the coefficient.

The coefficient of the first term in a polynomial is the lead coefficient.

A polynomial with two terms is called a binomial. A polynomial with three term is called a trinomial.

Polynomials in one variable

The degree of a polynomial in one variable is the largest exponent of that variable.

1425 2 −+ xx

14 +x

A constant has no variable. It is a 0 degree polynomial.2This is a 1st degree polynomial. 1st degree polynomials are linear.

This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.

183 3 −x This is a 3rd degree polynomial. 3rd degree polynomials are cubic.

For example:f(x) = 7, g(x) = -3/2, h(x) = 2

are constant polynomials.The degree of constant polynomials is not defined.

For example: p(x) = 4x – 3, q(x) = 3y are linear polynomials.Any linear polynomial is in the form ax + b, where a, b are real nos. and a ≠ 0.

It may be a monomial or a binomial. F(x) = 2x – 3 is binomial whereas g (x) = 7x is monomial.

A polynomial of degree two is called a quadratic polynomial.

f(x) = √3x2 – 4/3x + ½, q(w) = 2/3w2 + 4 are quadratic polynomials with real coefficients.

Any quadratic is always in the form f(x) = ax2 + bx +c where a,b,c are real nos. and a ≠ 0.

A polynomial of degree three is called a cubic polynomial.

f(x) = 9/5x3 – 2x2 + 7/3x _1/5 is a cubic polynomial in variable

x.

Any cubic polynomial is always in the form f(x = ax3 + bx2 +cx + d where a,b,c,d are

real nos.

Examples

Polynomials Degree Classify by degree

Classify by no. of terms.

5 0 Constant Monomial

2x - 4 1 Linear Binomial

3x2 + x 2 Quadratic Binomial

x3 - 4x2 + 1 3 Cubic Trinomial

A real no. x is a zero of the polynomial f(x), is f(x) = 0 Finding a zero of the polynomial means solving polynomial equation f(x) = 0.

If f(x) is a polynomial and y is any real no. then real no. obtained by replacing x by y in f(x) is cal led the value of f(x) at x = y and is denoted by f(x).

Value of f(x) at x = 1 f(x) = 2x 2 – 3x – 2 f(1) = 2(1) 2 – 3 x 1 – 2 = 2 – 3 – 2 = -3

Zero of the polynomial f(x) = x 2 + 7x +12 f(x) = 0 x2 + 7x + 12 = 0 (x + 4) (x + 3) = 0 x + 4 = 0 or, x + 3 = 0x = -4 , -3

• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In this case, a is also called a root of the equation p(x)=0.

• Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.

Number of TurnsAnother fact, that you’ll justify in Calculus I is that the graph of a polynomial of degree n can have at most n – 1 turns in it (where we switch from increasing to decreasing or back). The number of turns may be smaller than n – 1, but only by an even number.

Degree 1 0 turns Degree 2 1 turn

GENERAL SHAPES OF POLYNOMIAL

FUNCTIONS f(x) = 3

CONSTANT FUNCTION

DEGREE = 0

MAX. ZEROES = 0

GENERAL SHAPES OF POLYNOMIAL

FUNCTIONS f(x) = x + 2

LINEAR FUNCTION

DEGREE =1

MAX. ZEROES = 1

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x2 + 3x + 2

QUADRATIC FUNCTION

DEGREE = 2

MAX. ZEROES = 2

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x3 + 4x2 + 2

CUBIC FUNCTION

DEGREE = 3

MAX. ZEROES = 3

POLYNOMIAL FUNCTIONSGENERAL SHAPES OF POLYNOMIAL FUNCTIONSf(x) = x4 + 4x3 – 2x – 1 QuaDrticFunctionDegree = 4Max. Zeros: 4

POLYNOMIAL FUNCTIONSGENERAL SHAPES OF POLYNOMIAL FUNCTIONSf(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 QuinticFunctionDegree = 5Max. Zeros: 5

Even More Graphs

0 turns

Degree 5

or 2 turns or 4 turns

1 turn

Degree 6

or 3 turns or 5 turns

Typical graph of a linear polynomial : y= 2x + 3

DPlotTrialVersionhttp://www.dplot.com

Polynomials1.grf

y=2x+3

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3

4

5

6

7

8

9

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3

4

5

6

7

8

9

Y-axis where all the x coordinates are 0

X-axis where all the y coordinates are 0

(-1.5,0)i.e.

x=-1.5; y=0

(0,3)i.e. x=0; y=3

0,0

Quadratic polynomial: ax2+bx+c

Y=x2-3x-4Y=x2-2x-3

These curves are called parabolas

Intercept in plain english means “cut”

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-8

-4

0

4

8

12

16

20

24

28

32

36

40

44

48

52

No intercept on x-axis

Types of quadratic polynomials

Graph a) has two real Zeros.  It has two x-intercepts.Graph b) has no real zeros.  It has no x-intercepts.  Both zeros are complex.

Graph c) the intercept shows only one zero. Though it is a polynomial of two degree it has only one zero!

How many

Zeros? in this

polynomial

No Zero in this case it does not

intercept the x axis

Two zeros as the curve

intercepts at two points on the x =axis

The maximum zeros the second degree polynomial has is TWO

Cubic Polynomial: p(x)=x3-4x

DPlotTrialVersionhttp://www.dplot.com

Plot2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Points where the curve interceptsThe x – axis:

The zeros are: -2.0,2

Thus the third degree polynomial has three zeros( roots)

Behavior at Zeros:

QUADRATICQUADRATIC

☻ α + β = - coefficient of x

Coefficient of x2

= - ba

☻ αβ = constant termCoefficient of x2

= ca

CUBICCUBIC

α + β + γ = -Coefficient of x2 = -bCoefficient of x3 a

αβ + βγ + γα = Coefficient of x = cCoefficient of x3 a

αβγ = - Constant term = dCoefficient of x3 a

)(*)(x 2 POZxSOZ +−

ON VERYFYI

NG THE

RELATIONSHIP B

ETWEEN

THE ZEROES A

ND

COEFFICIEN

TS

ON FINDING THE

VALUES OF EXPRESSIONS

INVOLVING ZEROES OF

QUADRATIC POLYNOMIAL

ON FINDING AN

UNKNOWN WHEN A

RELATION BETWEEEN

ZEROES AND COEFFICIENTS

ARE GIVEN.

OF ITS A QUADRATIC

POLYNOMIAL WHEN

THE SUM AND

PRODUCT OF ITS

ZEROES ARE GIVEN.

If f(x) and g(x) are any two polynomials with g(x) ≠ 0,then we can always f ind polynomials q(x), and r(x) such that :

F(x) = q(x) g(x) + F(x) = q(x) g(x) + r(x),r(x) ,

Where r(x) = 0 or degree r(x) < degree g(x)

ON VERYFYING THE DIVISION ALGORITHM FOR POLYNOMIALS.

ON FINDING THE QUOTIENT AND REMAINDER USING DIVISION ALGORITHM.

ON CHECKING WHETHER A GIVEN POLYNOMIAL IS A FACTOR OF THE OTHER POLYNIMIAL BY APPLYING THEDIVISION ALGORITHM

ON FINDING THE REMAINING ZEROES OF A POLYNOMIAL WHEN SOME OF ITS ZEROES ARE GIVEN.

F(x) = P(x) G(x) + R(x)

D. Theorm Remainder

If polynomials F(x) = P(x) H(x) + R and P(x) = (x-h), so

we will get : F(x) = (x-h). H(x) + RTHEORM : if the polynomials F(x) is divided by

(x-h), the remainder is F(h)Prove: F(x) = (x-h) . H(x) + R

F(h) = (h-h) . H(x) + R = 0 + R

F(h) = R

G. Factor Theorm

Theorm :If F(x) a polynomials then F(h) = 0

If and only if (x-h) is factor of F(x)

I. If F(h) =0 then (x-h) is factor of F(x)Prove:

F(x) = P(x) H(x) + R= (x-h) H(x) + R

F(h) = 0 = (x-h) H(x) + 0= (x-h) H(x)

Proved (x-h) is factor F(x)

H. Determined factor of Polynomials

Algorithm :1. If the sum coefficient xodd = the sum of

coefficient xeven so x = 1 is the factor of polynomials.

2. If the sum of coefficient xodd = sum of coefficient xeven so x=-1 is the factor of

polynomials.3. If the 1st and 2nd step not fulfill look at factor

of constant (by chance). Until find R of it is 0.

Algebraic Identities

Some common identities used to factorize polynomials

(x+a)(x+b)=x2+(a+b)x+ab(a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b)

Algebraic Identities

Advanced identities used to factorize polynomials

(x+y+z)2=x2+y2+z2+2xy+2yz+2zx

(x-y)3=x3-y3-3xy(x-y)

(x+y)3=x3+y3

+3xy(x+y)

x3+y3=(x+y) * (x2+y2-xy) x3-y3=(x+y)

* (x2+y2+xy)