2. DEFINITION
An algebraic expression that contain two or more terms is called a
polynomial.
Each term of a polynomial has a co-efficient.
For an algebraic expression to qualify for being a polynomial it is
necessary that the power of variable in each term must be a positive
integer.
e.g. –x3+4x2+7x-2, 3y2+5y-7 etc.
𝑥 +
1
𝑥
is not a polynomial as the power of x in second term is -1
𝑦+5y is not a polynomial as the power of y is fraction in first term.
3. A polynomial in variable can be expressed generally as
anxn +an-1xn-1+an-2xn-2+…………..+ a2x2+a1x+a0
where
an, an-1, an-2, …………., a2, a1, a0 are all constants not
all zero and n is any positive integer.
4. DEGREE OF A POLYNOMIAL
The degree of a polynomial is the highest power of the
variable appearing in the polynomial.
e.g.: The degree of the polynomial x2-6x +7 is 2 as the
highest power of x appearing in the given polynomial is 2.
3x5+4x2-2. Here the highest power of variable is 5 so the
degree is 5
5. CLASSIFICATION OF POLYNOMIALS:
S.No. DEGREE NAME OF POLYNOMIAL
1 Zero (0) Constant Polynomial
2 One (1) Linear Polynomial
3 Two (2) Quadratic Polynomial
4 Three (3) Cubic Polynomial
5 Four (4) Bi-Quadratic Polynomial
The polynomials on the basis of their degree can be classified
as:
7. ZEROES OF A POLYNOMIAL:
The zero or root of a polynomial p(x) is the value of x at
which p(x) = 0.
If for x=a, p(a) = 0 then we say x=a is the zero or root of p(x).
E.g.: p(x) = x2- 5x+6 is a polynomial.
For x= 2, we get p(2) = (2)2 – 5(2) +6
= 4 -10 +6 = 10-10 =0
Hence x=2 is a root or zero of p(x).
8. GEOMETRICAL MEANING OF ZEROES OF A
POLYNOMIAL
If p(x) is a polynomial and we plot the graph of p(x) on a graph
sheet. The number of points of intersection of graph pf p(x) and
X –axis represent the number of zeroes or roots of p(x).
A linear polynomial graph meets X axis at only one point.
A quadratic polynomial graph meets the X – axis at maximum two
points.
A cubic polynomial graph meets the X – axis at maximum three
points.
9. GRAPH OF A LINEAR POLYNOMIAL
Here p(x) = 5x-15, meets the X axis at A (3,0).
So, x=3 is a root or zero of p(x).
Graph of a linear function meets X –axis
at maximum one point
10. GRAPH OF A QUADRATIC FUNCTION
Here p(x) = x2-x-6 meets X axis at A (-2,0) and B (3,0). So x= -2 and x=3 are the
roots of p(x).
The graph of a quadratic polynomial is a parabola
and meets X –axis at maximum two points
11. GRAPH OF A CUBIC FUNCTION
Here p(x) = x3-7x; meets X- axis at A(-2.6,0); B(0,0); C(2.6,0). So,
x= -2.6, 0 and +2.6 are the roots of p(x).
12. RELATIONSHIP BETWEEN ZEROES AND CO-
EFFICIENTS OF A POLYNOMIAL
If ax2+bx+c is a quadratic polynomial whose roots are α and β.
Then
Sum of roots α + β =
−𝒃
𝒂
=
−(𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙)
(𝒄𝒐−𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙𝟐)
Product of roots α X β =
c
a
=
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
(𝒄𝒐−𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙𝟐)
The quadratic polynomial is: x2 –(Sum) x + (Product)
13. If ax3+bx2+cx+d is a cubic polynomial whose roots
are α, β and γ, then
α + β + γ =
−𝒃
𝒂
=
− (𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙𝟐 )
(𝒄𝒐−𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙𝟑)
αβ + βγ + γα =
𝒄
𝒂
=
(𝒄𝒐−𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙)
(𝒄𝒐−𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙𝟑)
α X β X γ =
−𝒅
𝒂
=
−(𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 )
(𝒄𝒐−𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙𝟑)
14. FIND THE ZEROES AND VERIFY THE
RELATIONSHIP BETWEEN CO-EFFS. & ZEROES
(i) p(x) = x2- 2x- 8 Solving by splitting the
middle term
= x2 -4x+2x-8 now regrouping and taking
out common factors
= x(x-4) + 2 (x-4)
= (x-4) (x+2)
For p(x) = 0 ; Either x-4 = 0 ⇒ x=4
or x+2 = 0 ⇒ x = -2
Hence zeroes are α = 4 and β = -2
Now the coefficients are
a = 1; b = -2 and c = -8
The sum of roots α + β = 4 +(-2) = 4-2 =
2
−𝑏
𝑎
=
−(−2)
1
= 2
Hence α + β =
−𝑏
𝑎
Product of roots = α X β = 4 X (-2) = -8
𝑐
𝑎
=
(−8)
1
= -8
Hence α X β =
𝑐
𝑎
Hence relation is verified.
15. (ii) 6x2 – 3 -7x
First arranging the terms in descending
order of power
6x2- 7x – 3 Solving by splitting the middle
term
= 6x2 – 9x+ 2x -3 Regrouping and taking out
common factor
= 3x(2x-3) +1 (2x-3)
= (2x-3) (3x+1)
For p(x) = 0
Either 2x-3 = 0 ⇒ 2x= 3 ⇒ x= 3/2
Or 3x+1 = 0 ⇒ 3x = -1 ⇒ x = -1/3
Hence zeroes are α = 3/2 and β = -1/3
Now the coefficients are
a = 6; b = -7 and c = -3
The sum of roots α + β =
3
2
+ (
−1
3
)
=
9−2
6
=
7
6
−𝑏
𝑎
=
−(−7)
6
=
7
6
The product of the roots α X β =
3
2
X (
−1
3
) =
−1
2
𝑐
𝑎
=
−3
6
=
−1
2
Hence relation is verified.
16. PRACTICE QUESTIONS:
Q. Find a polynomial sum and product of whose roots are 4 and 1
respectively.
Solution: Here Sum (S) = 4 and product (P) = 1
Polynomial = x2- (Sum) x + Product
= x2 – 4x +1
Q. Find a polynomial whose roots are 2 and 5.
Solution: Here α = 2 and β = 5
Sum = α + β = 2 + 5 = 7
Product = α X β = 2 X 5 = 10
Polynomial = x2- (Sum) x + Product
= x2 – 7x +10
17. DIVISION ALGORITHM FOR POLYNOMIALS
If p(x) and g(x) are any two polynomials with g(x) ≠
𝟎 then we can find polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x) where r(x) = 0 or
degree of r(x) < degree of g(x).
Here again, we see that
Dividend = Divisor × Quotient + Remainder
18. Divide: 3x2-x3-3x+5 by x-1-x2
Here Divisor is x-1-x2 = -x2 + x -1
and Dividend = 3x2– x3 -3x +5 = -x3 +3x2 – 3x + 5
