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

Eg.   p(x) =3x5 +3x2 3x+5    has degree of 5   .

Linear polynomial:The polynomial with degree  1.

Eg.   P(X)=2x+1 ,3x+5, 2x5 ,3x

Quadratic polynomial:The polynomial with degree  2.

Eg.    p(x)= 2x2 +2x+1 ,3x2 +x+5 ,2x24 , 3x2

Cubic polynomial:The polynomial with degree 3 .

Eg.    p(x) =2x3+2x+1 ,3x3 +x2+1 ,2x35,5x3

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

eg . p(x) =2x2 +2x+1 ,
p(2) = 2(2)2 +2(2) +1 =5

value of p(x) at  x=-2 =p (-2) =5

Zero/root of polynomial:   If   p(k) = 0

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

Geometrical meaning of zero:

Linear polynomial:For a linear polynomial  p(x) =ax+b
(b/a)  is zero of the polynomial.
And graph of p(x)  cuts   axis at only one point  (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


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

 To prove:  p  divides a  , i.e p=ka  , for some integer 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.

  1. 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)  Quadratic polynomial:  ax2+bx+c
If α ,β  are two roots of quadratic polynomial.
sum of zeroes  = coefficient of Xcoefficient of x2
i.e. α+β = ba
product of zeroes =coefficient termscoefficient of x2
i.e. αβ =ca
2)Cubic polynomial:  p(x) = ax3+bx2+cx+d
Ifα,β,γ  are two roots of cubic polynomial.
sum of zeroes  =coefficient of x2coefficient of x3
α+β+γ = ba
sum of product  taken two at a time  =coefficiant of xcoefficiant of x3
i.e.  αβ+βγ+αγ  =ca  
product of zeroes =constant termscoefficient of x3
i.e.product of zeroes =αβ=da

Division algorithm for polynomial:

p(x) and g(x)  are two polynomials with degree of
g(x)<degree of p(x) then p(x)=q(x)g(x)+r(x) ,where r(x)=0  or 
 deg r(x) deg g(x)

Or we can express polynomial as

   Dividend =Quotient × Divisor +Remainder

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