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1. B.J.P.S Samiti’s
M.V.HERWADKAR ENGLISH MEDIUM HIGH SCHOOL
CLASS 10th: POLYNOMIALS
Program:
Semester:
Course: NAME OF THE COURSE
Staff Name: VINAYAK PATIL 1

2. POLYNOMIALS
•Any expression containing one or more
terms with non zero coefficients (and
variables with non negative integers as
exponents) is called a polynomials.
•For Example: 10, a + b, 7x + y + 5, w + x +
y + z, etc. are some polynomials
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 2

3. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 3
DEGREE OF A POLYNOMIAL
• The highest power of the variable in a
polynomial is the degree of the polynomial.
• For example, 4x + 2 is a polynomial in the
variable x of degree 1,
• The degree of a non-zero constant polynomial
is zero.

4. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 4
•A polynomial of one term is called a monomial.
•Examples: 2x, 3xyz, -5, ¾ z etc
•A polynomial of two terms is called a binomial.
•Examples: 5y-3, 4z³+7, 5xyz –x etc
•A polynomial of three terms is called. a trinomial
•Examples:90xz+16x -¼, x-y-7, 2ax +3by –xy etc
TYPES OF POLYNOMIAL

5. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 5
• A polynomial of degree one is called a linear polynomial.
• Examples: 3x-5, 8x+7y-9z, ½ x-6z-10√2 etc
• A polynomial of degree two is called a quadratic
polynomial.
• Examples: 3x²-5x+4, 3x²+7y-9z, ½ 6z²+3x-9 etc
• A polynomial of degree three is called a cubic polynomial
• Examples: x³+13x²-5x+14, 8xy+7y-9z³, x-6z²-8y³
TYPES OF POLYNOMIALS (DEGREE)

6. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 6
• 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.
• In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e.,
k=
−𝒃
𝒂
• So, the zero of the linear polynomial ax + b is
−𝒃
𝒂
=
(Constant term)
Coefficient of x
ZERO OF A POLYNOMIAL

7. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 7
Geometrical Meaning of the Zeroes of
a Polynomial
In general, for a linear
polynomial ax + b, a ≠ 0, the
graph of y = ax + b is a straight
line which intersects the x-axis at
exactly one point, namely, (0
−𝒃
𝒂
).
Therefore, the linear polynomial
ax + b, a 0, has exactly one zero,

8. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL
8
Geometrical Meaning of the Zeroes of
a Polynomial
For any quadratic polynomial ax² + bx + c, a ≠ 0, the
graph of the corresponding equation y = ax² + bx + c
has one of the two shapes U either open upwards or
open downwards depending on whether a > 0 or a < 0.
These curves are called parabolas.
A parabola is a plane curve which is mirror
symmetrical and approximately U-shaped.
The zeroes of a quadratic polynomial ax² + bx + c, a ≠
0, are precisely the x-coordinates of the points where
the parabola representing y = ax² + bx + c intersects
the x-axis. We can see geometrically, from the
following graphs, that a quadratic polynomial can
have either two distinct zeroes or two equal zeroes (i.e.,
one zero), or no zero. This also means that a
polynomial of degree 2 has at most two zeroes

9. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 9
•In general, given a polynomial p(x)
of degree n, the graph of y = p(x)
intersects the x-axis at at most n
points. Therefore, a polynomial
p(x) of degree n has at most n
zeroes.

10. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 10
EXERCISE 9.1
1. The graphs of y =
p(x) are given in Fig.
9.10 below, for some
polynomials p(x). Find
the number of zeroes of
p(x), in each case

11. RELATIONSHIP BETWEEN ZEROES AND
COEFFICIENTS OF A POLYNOMIAL
• In general, if α and β are the zeroes of the quadratic polynomial
• p(x) = ax² + bx + c, a ≠ 0 then x – α and x –β are the factors of
p(x).
• Sum of its zeroes = α + β =
−𝒃
𝒂
= − (coefficient of x)/(coefficient
ofx²)
• Product of its zeroes = α × β =
𝒄
𝒂
(Constant term)/(coefficient of
x²)
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 11

12. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 12
EXERCISE 9.2
1. Find the zeroes of the following quadratic polynomials
and verify the relationship between the zeroes and the
coefficients.
(i) 𝒙𝟐
– 2x – 8 (ii) 4𝒔𝟐
– 4s + 1
2. Find a quadratic polynomial each with the given
numbers as the sum and product of its zeroes respectively
(iv) 1, 1 (vi) 4, 1

13. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 13
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).
This result is known as the Division Algorithm for
polynomials

14. • Example: Divide.x³ – 4x² + 2 x -3 by x+2
• So, here the quotient is x² – 6x + 14 and the
remainder is − 31. Also,
• (x+2)(x² –6x+ 14) + (−31 )
• =x³− 6x² +14x +2x² −12x +28 −31
• = x³ −4x² + 2x -3
• Therefore,
• Dividend = Divisor × Quotient + Remainder
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL

15. EXERCISE 9.3
•1. Divide the polynomial p(x) by the polynomial
g(x) and find the quotient and remainder in each
of the following :
(i) p(x) = 𝒙𝟑
– 3𝒙𝟐
+ 5x – 3, g(x) = 𝒙𝟐
– 2
(ii) p(x) = 𝒙𝟒
– 3𝒙𝟐
+ 4x + 5, g(x) = 𝒙𝟐
+ 1 – x
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 15

16. EXERCISE 9.3
3) Obtain all other zeroes of 3𝒙𝟒
+ 6 𝒙𝟑
– 2 𝒙𝟐
– 10x – 5,
if two of its zeroes are
𝟓
𝟑
and −
𝟓
𝟑
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 16

17. EXERCISE 9.3
•4. On dividing 𝒙𝟑 – 3 𝒙𝟐 + x + 2 by a
polynomial g(x), the quotient and
remainder were x – 2 and –2x + 4,
respectively. Find g(x).
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 17

18. HOMEWORK
1)Find the zeros of the polynomial p(x) =x² + x -6 and
verify the relationship between zeros and coefficients.
2) Divide 𝒙𝟒
+ 6x³ - 3x² + 3x + 5 by x -2 and verify
division algorithm for polynomials.
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 18