Polynomial

By Muntaha Sheikh

1. An Assignment on “Polynomial”
Gaur International School
Submitted By:
Muntaha Sheikh
IX-B (Mathematics)

2. Content
●An introduction of Polynomials
● Polynomials in one variable
●Degree of polynomial
●Types of polynomial
●Zeros of a polynomial
●Remainder Theorem
●Algebraic Identities

3. Introduction
●Polynomial is a single term or a sum of a finite
number of terms.
●In mathematics : a polynomial is an expression
consisting of variables (or indeterminate) and
coefficients, that involves only the operations of
addition, subtraction, multiplication, and non-
negative integer exponents

4. Polynomials in one variable
● A polynomial in one variable X is an algebraic
expression in X of the form.
● NOT A POLYNOMIAL :
The expressions like 1÷x − 1,∫x+2 etc are not
polynomials

5. Degree of polynomial
●The highest power of x in p(x) is called the degree
of the polynomial p(x)
●For example
1) p(x) = 3x +½ is a polynomial in the variable x of
degree 1
2) q(y) = 2y² − ⅜ y +7 is a polynomial in the
variable y of degree 2

6. Types of polynomial
●Constant polynomial
●Linear polynomial
●Quadratic polynomial
●Cubic polynomial
●Bi-quadratic polynomial

7. Constant polynomial
●A polynomial of degree zero is called a constant
polynomial
For example: p(x) = 7 etc
It is also called zero polynomial
The degree of the zero polynomial is not defined

8. Linear polynomial
●A polynomial of degree 1 is called a linear
polynomial
●For example:
p(x)=2x−3 , q(x)=3x +5 etc
The most general form of a linear polynomial is
ax + b , a ≠ 0 ,a & b are real.

9. Quadratic polynomial
●A polynomial of degree 2 is called quadratic
polynomial
●For example
2x² + 3x − ⅔ , y² − 2 etc
More generally , any quadratic polynomial in x with
real coefficient is of the form ax² + bx + c , where a,
b ,c, are real numbers and a ≠ 0

10. Cubic polynomial
● A polynomial of degree 3 is called a cubic
polynomial
●For example:
p(x)= 2 − x³ , x³, etc
The most general form of a cubic polynomial with
coefficients as real numbers is ax³ + bx² + cx + d ,
a ,b ,c ,d are real

11. Zeros of a polynomial
● A real number k is said to a zero of a polynomial
p(x), if p(k) = 0.
●For example:
consider the polynomial p(x) = x³ − 3x − 4 .
Then, p(−1) = (−1)² − (3(−1) − 4 = 0
Also, p(4) = (4)² − (3 ×4) − 4 = 0
Here, − 1 and 4 are called the zeroes of the quadratic
polynomial x² − 3x − 4 .

12. Remainder Theorem
●Let p(x) be any polynomial of degree greater than or
equal to one and let be any real number. If p(x) is
divided by the linear polynomial x-a then the
remainder is p(a).
●For example:
If p(x)=x
4
+x
3
-2x
2
+x+1 is divisible by x-1. Then
p(1)=(1)
4
+(1)
3
-2(1)
2
+1+1
= 2
Hence, p(x) is divisible by x-1 and remainder is 2

13. Algebraic Identities
●An algebraic identity is an algebraic equation that is
true for all values of the variables occurring in it.
●For example:
(X+Y)
2
= X
2
+2XY + Y
2
(X+a)(X+b) = X
2
+ (a+b)X + ab

14. Reference
'Mathematics' A text book for class IX, NCERT