Polynomials

Tutormate > CBSE Syllabus-Class 10th Maths > Polynomials

02 POLYNOMIALS

Degree of polynomial:The highest power of x in p(x) is called degree of the polynomial.

E g . p (x) = 3 x^{5} + 3 x^{2^{}} - 3 x + 5 h a s d e g r e e o f 5 .

Linear polynomial:The polynomial with degree 1.

E g . P (X) = 2 x + 1, 3 x + 5, \sqrt{2} x - 5, 3 x

Quadratic polynomial:The polynomial with degree 2.

E g . p (x) = 2 x^{2} + 2 x + 1, 3 x^{2} + x + 5, 2 x^{2} - \sqrt{4}, 3 x^{2}

Cubic polynomial:The polynomial with degree 3 .

E g . p (x) = 2 x^{3} + 2 x + 1, 3 x^{3} + x^{2^{}} + 1, 2 x^{3} - 5, 5 x^{3}

Value of polynomial: Value obtained by replacing by some real number k , is value of polynomial at x=k .It is denoted by .

e g . p (x) = 2 x^{2} + 2 x + 1,

p (- 2) = 2 (- 2)^{2} + 2 (- 2) + 1 = 5

value of p(x) at x=-2 =p (-2) =5

Z e r o / r o o t o f p o l y n o m i a l : I f p (k) = 0

for some k ,Kis called zero or root of polynomial.

Geometrical meaning of zero:

L i n e a r p o l y n o m i a l : F o r a l i n e a r p o l y n o m i a l p (x) = a x + b

p (- b / a) = a (- b / a) + b = 0

(- b / a) i s z e r o o f t h e p o l y n o m i a l .

A n d g r a p h o f p (x) c u t s a x i s a t o n l y o n e p o i n t (- b / a, 0)

So, we say linear polynomial p(x) has only one zero, and it is the point where its graph cuts x axis

i . e x = (- b / a)

Quadratic polynomial

p (x) = a x^{2} + b x + c

Case 1):a>0 ,. graphis upward parabola .

T o p r o v e : p d i v i d e s a, i . e p = k a, f o r s o m e i n t e g e r k .

Graph cuts x axis at 2 points or point or may not cut x axis.

Case 2 :a<0,The graph is downward parabola.

Graph cuts axis at points or point or may not cut x axis.

In either cases i) If the graph cuts x axis in two points, then we say polynomial has 2 zeroes.

ii) If the graph cuts x axis in only one point, then we say polynomial has 1 zero.

iii) If the graph does not cut x axis, then we say zeroes of polynomial do not exist.

Relation between zeroes and coefficients of polynomial:

1) Q u a d r a t i c p o l y n o m i a l : a x^{2} + b x + c

I f α, β a r e t w o r o o t s o f q u a d r a t i c p o l y n o m i a l .

s u m o f z e r o e s = \frac{- c o e f f i c i e n t o f X}{c o e f f i c i e n t o f x^{2}}

i . e . α + β = \frac{- b}{a}

p r o d u c t o f z e r o e s = \frac{c o e f f i c i e n t t e r m s}{c o e f f i c i e n t o f x^{2}}

i . e . α β = \frac{c}{a}

2) C u b i c p o l y n o m i a l : p (x) = a x^{3} + b x^{2} + c x + d

I f α, β, γ a r e t w o r o o t s o f c u b i c p o l y n o m i a l .

s u m o f z e r o e s = \frac{- c o e f f i c i e n t o f x^{2}}{c o e f f i c i e n t o f x^{3}}

α + β + γ = \frac{- b}{a}

s u m o f p r o d u c t t a k e n t w o a t a t i m e = \frac{c o e f f i c i a n t o f x}{c o e f f i c i a n t o f x^{3}}

i . e . α β + β γ + α γ = \frac{c}{a}

p r o d u c t o f z e r o e s = \frac{- c o n s \tan t t e r m s}{c o e f f i c i e n t o f x^{3}}

i . e . p r o d u c t o f z e r o e s = α β = \frac{- d}{a}

Division algorithm for polynomial:

p (x) a n d g (x) a r e t w o p o l y n o m i a l s w i t h d e g r e e o f

g (x) < d e g r e e o f p (x) t h e n p (x) = q (x) g (x) + r (x), w h e r e r (x) = 0 o r

d e g r (x) \leq d e g g (x)

Or we can express polynomial as

D i v i d e n d = Q u o t i e n t \times D i v i s o r + R e m a i n d e r

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Pair of linear equation

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Arithmetic Progression

Triangles

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Introduction to trigonometry

Application of trigonometric ratio

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Areas related to circle

Surface areas and volumes

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