Let's think step by step: 1) For x2 + 2x + 5 to be a factor of x4 + px2 + q, when we divide x4 + px2 + q by x2 + 2x + 5, the remainder should be 0. 2) Dividing x4 + px2 + q by x2 + 2x + 5, the quotient will be x2 and the remainder will be p - 5. 3) For the remainder to be 0, we must have p - 5 = 0. Therefore, p = 5 4) Putting p = 5 in the original expression, we get: x4 + 5x2 + q 5) Comparing the constant

1. POLYNOMIALS
FOR CLASS 10
PREPARED BY
GOLAM ROBBANI AHMED

2. a polynomial is
an expression consisting
of variables and coefficients, that
involves only the operations
of addition, subtraction, multiplication,
and non-negative integer as
exponents of variables.
An example of a polynomial of a single
variable x is x2 − 4x + 7.
An example of Polynomial in three
variables is x3 + 2xyz2 − yz + 1

3. What is Variable?
A variable is a quantity that may change
within the context of a mathematical
problem or experiment.
Typically, we use a single letter to
represent a variable. The letters x, y,
and z are common generic symbols used
for variables.
Sometimes, we will choose a letter that
reminds us of the quantity it represents,
such as t for time, v for voltage etc.

5. Based on the number of terms of the given
polynomial, it can be divided
into monomial, binomial, trinomial,
constant polynomials. The polynomial with
only one term is called monomial.
The polynomial with two terms is
called binomial.
The polynomial with three terms are
called trinomial.

7. Based on the non negative integer exponent of Variable

9. Polynomial Degree Name using
Degree
Number of
Terms
Name using
number of
terms
7x + 4 1 Linear 2 Binomial
3x2
+ 2x + 1 2 Quadratic 3 Trinomial
4x3
3 Cubic 1 Monomial
9x4
+ 11x 4 Fourth degree 2 Binomial
5 0 Constant 1 monomial

10. FOLLOWINGS ARE NOT POLYNOMIAL

11. Standard form of Polynomial:
The polynomial above is in standard form. Standard form
of a polynomial - means that the degrees of its monomial
terms decrease from left to right.

12. The degree of a polynomial:
The degree of a polynomial in one variable is the largest
exponent in the polynomial.
Ex:
5x12−2x6+x5−198x+1 degree :12
X4−x3+x2−x+1 degree :4
-8 degree: 0
On the otherhand Polynomials in two variables are algebraic
expressions consisting of terms in the form axnym
The degree of each term in a polynomial in two variables is the
sum of the exponents in each term and the degree is the highest
sum.

17. Relationship between the zeros and coefficients of a
quadratic polynomial?

20. Q. If the squared difference of the zeros of the quadratic
polynomial x² + px + 45 is equal to 144 , find the value of p.
Ans: Let two zeros are α and β where α > β
According given condition
(α - β)2 = 144
Let p(x) = x² + px + 45
α + β = − p
αβ = 45
now (α + β)² = (α - β)² + 4 αβ
(-P)² = 144 +180=324
Solving this we get p = ± 18

22. Question :
If two zeroes of the polynomial are find other
zeros
= x² + 4 − 4x − 3
= x² − 4x + 1 is a factor of the given
polynomial
For finding the remaining zeroes of the given
polynomial, we will find the quotient by
dividing

23. = =
Or x = 7 or −5
Hence, 7 and −5 are also zeroes of this polynomial.

25. Q. if p(x) a Polynomial and p(a)p(b)<0 then find number of zeroes of the
polynomial between a and b.
Here
p(a)p(b)<0
i.e. if P(a) is positive then P(b) is negative and vice
versa.
This will be possible when P(x) is either linear, cubic,and
so on
i.e zeroes will be 1,3,5,………………….

26. Q.What will be the value of p and q for x2 + 2x + 5 to
be a factor of x4 + px2 + q