POLYNOMIAL IN ONE POLYNOMIAL IN ONE VARIABLEVARIABLE

For real variable x, an expression of the type For real variable x, an expression of the type P (x) = aP (x) = a00 + a + a11x + ax + a2 2 xx22 + ----- + a + ----- + an n x x nn with real with real

co-efficient aco-efficient a0,0, a a11, a, a2, 2, aan n and n is positive and n is positive

integer is the real polynomial in one variable.integer is the real polynomial in one variable.

POLYNOMIAL EQUATIONS POLYNOMIAL EQUATIONS

If If P (x) is a real polynomial then P (x) = 0 is a P (x) is a real polynomial then P (x) = 0 is a polynomial equation.polynomial equation.

2x + 3 is a polynomial and 2x + 3 = 0 is 2x + 3 is a polynomial and 2x + 3 = 0 is polynomial equation.polynomial equation.

2x 2x -1-1 + 3 is not a polynomial but 2x + 3 is not a polynomial but 2x -1-1 + 3 = 0 + 3 = 0 is still a polynomial equation.is still a polynomial equation.

ZERO OF POLYNOMIALZERO OF POLYNOMIAL

The value of variable x = a is The value of variable x = a is called the zero of polynomial p called the zero of polynomial p (x) if P (a) = 0.(x) if P (a) = 0.

ZERO OF POLYNOMIALZERO OF POLYNOMIAL

The zero of an polynomial are the x The zero of an polynomial are the x co-ordinates (Abscissa) of the points co-ordinates (Abscissa) of the points where curve y = f (x) crosses the x-where curve y = f (x) crosses the x-axis.axis.

e.g. for x + 1 consider y = x + 1e.g. for x + 1 consider y = x + 1

o-1

1

X

Y/

X = -1 is the zero of the polynomial x + 1.

ZERO OF POLYNOMIALZERO OF POLYNOMIAL

For polynomial xFor polynomial x22 -3x + 2 -3x + 2 Consider y = xConsider y = x22 -3x + 2 -3x + 2

o 1X

Y/

The zeros of the polynomial are 1 and 2.

2

TYPES OF POLYNOMIALSTYPES OF POLYNOMIALS

Polynomials are named based on two Polynomials are named based on two criterions criterions

1. number of terms the polynomial 1. number of terms the polynomial has.has.

2. the highest exponent of the 2. the highest exponent of the variable present in the polynomial.variable present in the polynomial.

TYPES OF POLYNOMIALSTYPES OF POLYNOMIALS

Polynomial

Number of terms Highest Exponent of the variable

Monomial

Trinomial

Binomial

Linear

Cubic

Quadratic

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF LINEAR POLYNOMIALOF LINEAR POLYNOMIAL

For linear polynomial a x + b , a For linear polynomial a x + b , a ≠ 0, consider ≠ 0, consider y = a x + b.y = a x + b.

Linear polynomial represent a straight line Linear polynomial represent a straight line intersecting x-axis at point (-b/a, 0) and y-axis intersecting x-axis at point (-b/a, 0) and y-axis at (0,b)at (0,b)For - 2x + 4, curve is a straight line interesting x-axis at (2,0) and y-axis at (0,4)

4

2

X

Y

O

QuadraticQuadratic PolynomialPolynomial

Polynomial in one variable is called Polynomial in one variable is called quadratic quadratic polynomial if apolynomial if a33, a , a 44 ,------ = 0 ,------ = 0 , a , a 22 ≠≠ 0 and a 0 and a 00, , aa 1 1 may or may not may or may not be zerobe zero. .

In general, a quadratic polynomial is In general, a quadratic polynomial is denoted asdenoted as a x a x22 + b x + c , with a + b x + c , with a ≠≠ 0 0

XX22 –5x +4 –5x +4

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

a

cx

a

bxay 2

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

2

22

42 a

b

a

c

a

bxay

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

2

22

4

4

2 a

bac

a

bxay

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

a

acby

a

bxa

4

4

2

22

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

a

Dy

a

bxa

42

2

Put b2 – 4ac = D

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

a

Dy

a

bxa

42

2

It is a parabola with vertex at (-b/2a,-D/4a)

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

a

Dy

a

bxa

42

2

If a > 0

,2 4

b D

a a

0,

2a

b No Zeros

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION OF QUADRATIC POLYNOMIALOF QUADRATIC POLYNOMIAL

For quadratic polynomial a xFor quadratic polynomial a x22 + b x + c , a + b x + c , a ≠ 0, ≠ 0, consider y = consider y = a xa x22 + b x + c. + b x + c.

a

Dy

a

bxa

42

2

If a < 0

,2 4

b D

a a

0,

2a

b No Zeros

SIGN OF QUADRATIC SIGN OF QUADRATIC POLYNOMIALPOLYNOMIAL

CASE ICASE I

For D = b For D = b 22 – 4 ac < 0 – 4 ac < 0

If a > 0 then f (x) > 0 for all real values of x.If a > 0 then f (x) > 0 for all real values of x.

If a < 0 then f (x) < 0 for all real values of x.If a < 0 then f (x) < 0 for all real values of x.

FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0

a > 0, f (x) > 0a > 0, f (x) > 0

a < 0, f (x) < 0a < 0, f (x) < 0

SIGN OF QUADRATIC SIGN OF QUADRATIC POLYNOMIALPOLYNOMIAL

CASE IICASE II

For D = b For D = b 22 – 4 ac = 0 – 4 ac = 0

If a > 0 then f (x) ≥ 0 for all real values of x.If a > 0 then f (x) ≥ 0 for all real values of x.

If a < 0 then f (x) ≤ 0 for all real values of x.If a < 0 then f (x) ≤ 0 for all real values of x.

FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0

a > 0, f (x) ≥ 0a > 0, f (x) ≥ 0

a < 0, f (x) ≤ 0a < 0, f (x) ≤ 0

SIGN OF QUADRATIC SIGN OF QUADRATIC POLYNOMIALPOLYNOMIAL

CASE IIICASE III

For D = b For D = b 22 – 4 ac > 0 – 4 ac > 0

If a > 0 then f (x) ≥ 0 for all real values of x.If a > 0 then f (x) ≥ 0 for all real values of x.

If a < 0 then f (x) ≤ 0 for all real values of x.If a < 0 then f (x) ≤ 0 for all real values of x.

FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0

a > 0, f (x) ≥ 0a > 0, f (x) ≥ 0

a < 0, f (x) ≤ 0a < 0, f (x) ≤ 0

SIGN OF QUADRATIC SIGN OF QUADRATIC POLYNOMIALPOLYNOMIAL

CASE IIICASE III

For D = b For D = b 22 – 4 ac > 0 – 4 ac > 0

If a > 0 then If a > 0 then

FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0

> 0 for x < α , x > β

f (x) = = 0 for x = α , β

< 0 for α < x > β

α βX

Y

SIGN OF QUADRATIC SIGN OF QUADRATIC POLYNOMIALPOLYNOMIAL

CASE IIICASE III

For D = b For D = b 22 – 4 ac > 0 – 4 ac > 0

If a < 0 then f (x) If a < 0 then f (x)

FOR THE EXPRESSION F(X) = a x 2 + b x + c, a ≠ 0

< 0 for x < α , x > β

f (x) = = 0 for x = α , β

> 0 for α < x > βα β

X

Y

CONCLUSIONCONCLUSION

Polynomial f (x) = a xPolynomial f (x) = a x22 + b x + c has + b x + c has the same sign as that of “a” except the same sign as that of “a” except when zeros of quadratic polynomial when zeros of quadratic polynomial are real and distinct and x lies are real and distinct and x lies between them. between them.

Descartes' Rule Of SignsDescartes' Rule Of Signs

The maximum number of positive real zeros of The maximum number of positive real zeros of a polynomial f (x) is the number of changes of a polynomial f (x) is the number of changes of sign from positive to negative and vice versa sign from positive to negative and vice versa in f (x). in f (x).

e.g. in case of xe.g. in case of x33 – 2 x – 2 x2 2 – x + 2 [ – x + 2 [ = ( x - 1) (x - 2) ( x +1)]= ( x - 1) (x - 2) ( x +1)]

+ + - - - - + Here zeros are 1, 2 & -1 + Here zeros are 1, 2 & -1

11st st 2 2 nd nd Positive zeros Positive zeros

The maximum number of negative real zeros of The maximum number of negative real zeros of a a

polynomial f (x) is the number of changes of sign polynomial f (x) is the number of changes of sign from from

positive to negative and vice versa in f (- x). positive to negative and vice versa in f (- x).

e.g. in case of xe.g. in case of x33 – 2 x – 2 x2 2 – x + 2 [ – x + 2 [ = ( x - 1) (x - 2) ( x = ( x - 1) (x - 2) ( x +1)]+1)]

f(- x) = - f(- x) = - xx33 – 2 x – 2 x2 2 + x + 2 + x + 2 Here zeros are 1, 2 Here zeros are 1, 2 & -1& -1

-- ++ ++ + +

11st st Negative Negative zeros zeros

POSITIVE OR NEGATIVE POSITIVE OR NEGATIVE ZEROSZEROS

X3 +2X2-9X-18 Find positive and X3 +2X2-9X-18 Find positive and negative zeros.negative zeros.

If the remainder on division of If the remainder on division of x3+2x2+kx+3 by x-3 is 21,find the x3+2x2+kx+3 by x-3 is 21,find the quotient and the value of quotient and the value of k.Hence ,find the zeros of the cubic k.Hence ,find the zeros of the cubic polynomial x3+2x2+kx-18 Ans polynomial x3+2x2+kx-18 Ans (3,-2,-3)(3,-2,-3)

QUESTIONSQUESTIONS

Find k so that x2+2x+k is a factor of Find k so that x2+2x+k is a factor of 2x4+x3-14x2+5x+6.Also find all the 2x4+x3-14x2+5x+6.Also find all the zeros of the polynomials.(-3: -3,1,2,-1/2; zeros of the polynomials.(-3: -3,1,2,-1/2; 1,-3)1,-3)

Given that the zeros of the cubic Given that the zeros of the cubic polynomial x3-6x2+3x+10 are of the polynomial x3-6x2+3x+10 are of the form a,a+b,a+2b for some real numbers form a,a+b,a+2b for some real numbers a and b,find the values of a and b as well a and b,find the values of a and b as well as the zeros of the given polynomials.(-as the zeros of the given polynomials.(-1,2,5)1,2,5)