Polynomial for class 9

A Presentation for class IX students on the chapter Polynomial.

1. Polynomials
PREPARED BY
GOLAM ROBBANI AHMED

2. Polynomials: An algebraic expression in which the
variables involved have only non-negative integral
powers is called a polynomial.

4. Difference between Algebric Expression and Polynomial:
A polynomial is always going to be an algebraic expression,
but an algebraic expression doesn't always have to be a
polynomial.
(i)An algebraic expression is an expression with a variable in
it, and a polynomial is an expression with multiple terms with
variables in it.
Algebraic expression (not polynomial): 3b
Polynomial: 4x² + 3x -7
ii) An expression is not a polynomial if it has a negative
exponent or fractional exponent

5. Polynomial Can Have
•A polynomial can have:
Constants
Variables
Exponents
Coefficients

6. Degree of Polynomials
•The degree of a polynomial is the highest degree
for a term.
•For e.g.-
•The polynomial 3 − 5x+ 2x
5
− 7x
9
has degree 9.

7. Types Of Polynomial
•Polynomials classified by degree –
Degree Name Example
undefined
Zero 0
0 (Non-zero)Constant 1
1 Linear X+1
2 Quadratic X
2
+1
3 Cubic X
3
+1
4 Quartic(Biquadratic) X
4
+1
5 Quintic X
5
+1
6 Sextic X
6
+1
7 Septic X
7
+1
8 Octic X
8
+1
9 Nonic X
9
+1
10 Decic X
10
+1
100 Hectic X
100
+1

8. Linear Polynomials
•In a different usage to the above, a
polynomial of degree1 is said to be linear,
because the graph of a function of that form
is a line.
•For e.g.-
•2x+1
•11y +3

9. Quadratic Polynomials
•In mathematics, a quadratic polynomialor
quadraticis a polynomial of degree two, also called
second-order polynomial. That means the exponents
of the polynomial's variables are no larger than 2.
•For e.g.-
•x
2
− 4x+ 7 is a quadratic polynomial,
while x
3
− 4x+ 7 is not.

10. Cubic Polynomials
•Cubic polynomial is a polynomial of
having degree of polynomial no more than 3
or highest degree in the polynomial should
be 3and should not be more or less than 3.
•For e.g.-
• x
3
+ 11x= 9x
2
+ 55
• x
3
+ x
2
+10x = 20

11. Biquadratic Polynomials
Biquadratic polynomial is a polynomial of
having degree of polynomial is no more
than 4or highest degree in the polynomial
is not more or less than 4.
For e.g.-
4x
4
+ 5x
3
–x
2
+ x -1
9y
4
+ 56x
3
–6x
2
+ 9x + 2

12. Types Of Polynomial
•Polynomial can be classified by number of non-zero term
Number of non-
zero terms
Name Example
0 ZeroPolynomial 0
1 Monomial X
2
2 Binomial X
2
+1
3 Trinomial X
3
+X+1

13. Zero Polynomials
•The constant polynomial whose coefficients are
all equal to 0. The corresponding polynomial
function is the constant function with value 0, also
called the zero map. The degree of the zero
polynomial is undefined, but many authors
conventionally set it equal to -1 or ∞.

14. Monomial, Binomial & Trinomial
Monomial:-
A polynomial with one term.
E.g. -5x
3
, 8, and 4xy.
Binomial:-
A polynomial with two terms which are not like terms.
E.g. -2x–3, 3x
5
+8x
4
, and 2ab–6a
2
b
5
.
Trinomial:-
A polynomial with three terms which are not like terms.
E.g. -x
2
+ 2x -3, 3x
5
-8x
4
+ x
3
, and a
2
b + 13x + c.

15. Polynomial or Not?

16. Followings are not Polynomial

17. 3x
4
+ 5x
2
–7x+ 1
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.
term
term
termterm

Polynomial

Degree

Name using
Degree

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

18. State whether each expression is a
polynomial. If it is, identify it.
1) 7y -3x + 4 Trinomial
2) 10x
3
yz
2
Monomial
3) Not a polynomial2
5
7
2
y
y


19. The Degreeof a monomial is the sum of the exponents of
the variables or it is the highest power of one variable
polynomial.
1) 5x
2 Degree: 2
2)4a
4
b
3
c Degree: 8
3)-3 Degree: 0

20. Find the degree of x
5
–x
3
y
2
+ 4
1.0
2.2
3.3
4.5
5.10

21. 3) Put in ascending order in terms of y:
12x
2
y
3
-6x
3
y
2
+ 3y -2x
-2x + 3y -6x
3
y
2
+ 12x
2
y
3
4)Put in ascending order:
5a
3
-3 + 2a -a
2
-3 + 2a -a
2
+ 5a
3

22. Write in ascending order in terms of y:
x
4
–x
3
y
2
+ 4xy–2x
2
y
3
1.x
4
+ 4xy–x
3
y
2
–2x
2
y
3
2.–2x
2
y
3
–x
3
y
2
+ 4xy + x
4
3.x
4
–x
3
y
2
–2x
2
y
3
+ 4xy
4.4xy –2x
2
y
3
–x
3
y
2
+ x
4

29. Dividing Polynomials
Long division of polynomials is similar to long division of
whole numbers.
dividend =(quotient Xdivisor)+remainder
When you divide two polynomials you can check the answer
using the following:

30. Division algorithm for polynomials
Ifp(x)andg(x)areanytwopolynomialswith
g(x)≠0,thenwecanfindtwounique
polynomialsq(x)andr(x)suchthat
p(x) =g(x) xq(x) + r(x)
where r(x) = 0 or degree of r(x) < degree of g(x)

31. + 2 2 3 1
2
 xxx
Example: Divide x
2
+ 3x–2 by x–1 and check the answer.
x
x
2
+ x
2x–2
2x + 2
–4
remainder
Check:x
x
x
xx 
2
2

1.xxxx 
2
)1(
2.xxxxx 2)()3(
22

3.2
2
2 
x
x
xx
4.22)1(2  xx
5.4)22()22(  xx
6.
correct(x+ 2)
quotient
(x+ 1)
divisor
+ (–4)
remainder
= x
2
+ 3x–2
dividend
Answer: Quotient = x + 2 and Remainder = -4

32. Example: Divide 4x +2x
3
–1 by 2x–2 and check the answer. 1 4 0 2 2 2
23
 xxxx
Write the terms of the dividend in
descending order.2
3
2
2
x
x
x

1.
x
2232
22)22( xxxx 
2.
2x
3
–2x
22233
2)22(2 xxxx 
3.
2x
2
+ 4xx
x
x

2
2
2
4.
+ xxxxx 22)22(
2

5.
2x
2
–2xxxxxx 6)22()42(
22

6.
6x–13
2
6

x
x
7.
+ 366)22(3  xx
8.
6x–6r e m a in d e r5)66()16(  xx
9.
5
Check: (x
2
+ x + 3)(2x–2) + 5
=4x +2x
3
–1
Answer: Quotient = x
2
+ x +3
Remainder = 5
5
Since there is no x
2
term in the
dividend, add 0x
2
as a placeholder.

33. Division of
polynomials
33 6 5 2
2
 xxx
x
x
2
–2x
–3x+ 6
–3
–3x + 6
0
Answer: Quotient = x –3 Remainder = 0
Check: (x–2)(x–3) = x
2
–5x+ 6
Example: Divide x
2
–5x+ 6 by x–2.

34. Example: Divide x
3
+ 3x
2
–2x+ 2 by x+ 3 and check the answer. 2 2 3 3
23
 xxxx
x
2
x
3
+ 3x
2
0x
2
–2x
–2
–2x –6
8
Check: (x + 3)(x
2
–2) + 8
= x
3
+ 3x
2
–2x+ 2
Answer: Quotient = x
2
–2
Remainder = 8
+ 2
Note: the first subtraction
eliminated two terms from
the dividend.
Therefore, the quotient
skips a term.
+ 0x

35. Division of
polynomials
35
1. Can x -2 be the remainder on division of a
polynomial p(x) by x + 3 ?
Ans. No. Here the degree of both the remainder and the
divisor are one which is not possible because the
remainder is either zero or its degree is lower than that
of the degree of the divisor.