POLYNOMIALS

MADE BY : DEV YADAV
SCHOOL : K.V. N.S.G. MANESAR
CLASS : IX “B”
SUBJECT : MATHS
SUBMITTED TO : Ms Madhur Abhispa

 A monomial is a real number, a variable,
or a product of a real number and one or
more variables with whole number
exponents.
› Examples:
 The degree of a monomial in one
variable is the exponent of the variable.
2 3
5, , 3 , 4x wy x

 A polynomial is a monomial or a sum of
monomials.
› Example:
 The degree of a polynomial in one
variable is the greatest degree among its
monomial terms.
› Example:
2
3 2 5xy x+ −
2
4 7x x− − +

 A polynomial function is a polynomial of
the variable x.
› A polynomial function has distinguishing
“behaviors”
 The algebraic form tells us about the graph
 The graph tells us about the algebraic form

 The standard form of a polynomial
function arranges the terms by degree in
descending order
› Example:
3 2
( ) 4 3 5 2P x x x x= + + −

Polynomial
s
Kinds of Polynomials
(according to number of terms)
A polynomial with only one term is called a
monomial.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a
trinomial.

 Polynomials are classified by degree
and number of terms.
› Polynomials of degrees zero through five
have specific names and polynomials with
one through three terms also have specific
names.
Degree Name
0 Constant
1 Linear
2 Quadratic
3 Cubic
4 Quartic
5 Quintic
Number of
Terms
Name
1 Monomial
2 Binomial
3 Trinomial

Monomial : Algebric expression that consists only one
term is called monomial.
Binomial : Algebric expression that consists two terms
is called binomial.
Trinomial : Algebric expression that consists three
terms is called trinomial.
Polynomial : Algebric expression that consists many
terms is called polynomial.

 A polynomial is a function in the form
 is called the leading coefficient
 is the constant term
 n is the degree of the polynomial
f (x) = an xn
+ ...+ a1x + a0
an
a0

Polynomials Degree Classify by
degree
Classify by no.
of terms.
5 0 Constant Monomial
2x - 4 1 Linear Binomial
3x2
+ x 2 Quadratic Binomial
x3
- 4x2
+ 1 3 Cubic Trinomial

Standard form means that the terms of the
polynomial are placed in descending order,
from largest degree to smallest degree.
Polynomial in standard form:
2 x3
+ 5x2
– 4 x + 7
Degree
Constant termLeading coefficient
Polynomial
s

Phase 1Phase 1 Phase 2Phase 2
To rewrite a
polynomial in
standard form,
rearrange the
terms of the
polynomial
starting with the
largest degree
term and ending
with the lowest
degree term.
The leading coefficient,
the coefficient of the
first term in a
polynomial written in
standard form, should
be positive.
How to convert a polynomial into standard form?

745 24
−−+ xxx
x5+4
4x 2
x− 7−
Write the polynomials in standard form.
243
5572 xxxx ++−−
3
2x+4
x− 7−x5+2
5x+
)7552(1 234
−+++−− xxxx
3
2x−4
x 7+x5−2
5x−
Remember: The lead
coefficient should be
positive in standard form.
To do this, multiply the
polynomial by –1 using
the distributive property.

Write the polynomials in standard form and identify the polynomial by
degree and number of terms.
23
237 xx −−1.
2. xx 231 2
++

23
237 xx −−
23
237 xx −−
3
3x− 2
2x− 7+
( )7231 23
+−−− xx
723 23
−+ xx
This is a 3rd
degree, or cubic, trinomial.

xx 231 2
++
xx 231 2
++
2
3x x2+ 1+
This is a 2nd
degree, or quadratic, trinomial.

Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder
is r(x).
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1]
degree r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).

Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by
remainder theorem, when f(x) is divided by (x-a), then
the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore f(a)=0(given]
(x-a) is a factor of f(x).

 Let α, β and γ be the zeroes of the polynomial ax³ + bx² + cx + d
 Then, sum of zeroes(α+β+γ) = -b = -(coefficient of x²)
a coefficient of x³
αβ + βγ + αγ = c = coefficient of x
Product of zeroes (αβγ) = -d = -(constant term)

I) Find the zeroes of the polynomial x² + 7x + 12and verify the relation between the
zeroes and its coefficients.
f(x) = x² + 7x + 12
= x² + 4x + 3x + 12
=x(x +4) + 3(x + 4)
=(x + 4)(x + 3)
Therefore,zeroes of f(x) =x + 4 = 0, x +3 = 0 [ f(x) = 0]
x = -4, x = -3
Hence zeroes of f(x) are α = -4 and β = -3.

2) Find a quadratic polynomial whose
zeroes are 4, 1.
sum of zeroes,α + β = 4 +1 = 5 = -b/a
product of zeroes, αβ = 4 x 1 = 4 = c/a
therefore, a = 1, b = -4, c =1
as, polynomial = ax² + bx +c
= 1(x)² + { -4(x)} + 1
= x² - 4x + 1
The end

Some common identities used to factorize polynomials
(x+a)(x+b)=x2+(a+b)x+ab(a+b)2
=a2
+b2
+2ab (a-b)2
=a2
+b2
-2ab a2
-b2
=(a+b)(a-b)

Advanced identities used to factorize polynomials
(x+y+z)2
=x2
+y2
+z2
+2xy+2yz+2zx
(x-y)3
=x3
-y3
-
3xy(x-y)
(x+y)3
=x3
+y3
+
3xy(x+y)
x3
+y3
=(x+y) *
(x2
+y2
-xy) x3
-y3
=(x+y) *
(x2
+y2
+xy)

 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.
 Every linear polynomial in one variable has a unique
zero, a non-zero constant polynomial has no zero, and
every real number is a zero of the zero polynomial.

POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial
or a sum of monomials.
A POLYNOMIAL IN ONE
VARIABLE is a polynomial
that contains only one
variable.Example: 5x2
+ 3x - 7

A polynomial function is a function of the form
f(x) = an xn
+ an – 1 xn – 1
+· · ·+ a1 x + a0
Where an ≠ 0 and the exponents are all whole numbers.
A polynomial function is in standard form if its terms are
written in descending order of exponents from left to right.
For this polynomial function, an is the leading coefficient,
a0 is the constant term, and n is the degree.

The DEGREE of a polynomial in one
variable is the greatest exponent of its
variable.
A LEADING COEFFICIENT is the
coefficient of the term with the highest
degree.
What is the degree and leading
coefficient of 3x5
– 3x + 2 ?

A polynomial equation used to represent a
function is called a POLYNOMIAL
FUNCTION.
Polynomial functions with a degree of 1 are
called LINEAR POLYNOMIAL FUNCTIONS
called QUADRATIC POLYNOMIAL
FUNCTIONS
called CUBIC POLYNOMIAL FUNCTIONS

Degree Type Standard Form
You are already familiar with some types of polynomial
functions. Here is a summary of common types of
polynomial functions.
4 Quartic f (x) = a4x4
+ a3x3
+ a2x2
+ a1x + a0
0 Constant f (x) = a0
3 Cubic f (x) = a3x3
+ a2 x2
+ a1x + a0
2 Quadratic f (x) = a2 x2
+ a1x + a0
1 Linear f (x) = a1x + a0

The largest exponent within the polynomial
determines the degree of the polynomial.
Polynomial
Function in General
Form
Degree Name of
Function
1 Linear
2 Quadratic
3 Cubic
4 Quarticedxcxbxaxy ++++= 234
dcxbxaxy +++= 23
cbxaxy ++= 2
baxy +=

What is the degree of the monomial?
24
5 bx
The degree of a monomial is the sum of the exponents of the variables
in the monomial.
The exponents of each variable are 4 and 2. 4+2 = 6.
The degree of the monomial is 6.
The monomial can be referred to as a sixth degree monomial.

14 +x
83 3
−x
1425 2
−+ xx
The degree of a polynomial in one variable is the largest exponent of
that variable.
2 A constant has no variable. It is a 0 degree polynomial.
This is a 1st
degree polynomial. 1st
degree polynomials are linear.
This is a 2nd
degree polynomial. 2nd
degree
polynomials are quadratic.
This is a 3rd
degree polynomial. 3rd
degree polynomials are
cubic.

Classify the polynomials by degree and number of terms.
Polynomial
a.
b.
c.
d.
5
42 −x
xx +2
3
14 23
+− xx
Degree
Classify by
degree
Classify by
number of
terms
Zero Constant Monomial
First Linear Binomial
Second Quadratic Binomial
Third Cubic Trinomial

Find the sum. Write the answer in standard format.
(5x3
– x + 2 x2
+ 7) + (3x2
+ 7 – 4 x) + (4x2
– 8 – x3
)
Adding Polynomials
SOLUTION
Vertical format: Write each expression in standard form. Align like terms.
5x3
+ 2x2
– x + 7
3x2
– 4x + 7
– x3
+ 4x2
– 8+
4x3
+ 9x2
– 5x + 6

Find the sum. Write the answer in standard format.
(2 x2
+ x – 5) + (x + x2
+ 6)
Adding Polynomials
SOLUTION
Horizontal format: Add like terms.
(2x2
+ x – 5) + (x + x2
+ 6) = (2x2
+ x2
) + (x + x) + (–5 + 6)
= 3x2
+ 2x + 1

Find the difference.
(–2x3
+ 5x2
– x + 8) – (–2x2
+ 3x – 4)
Subtracting Polynomials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you
multiply each term in the subtracted polynomial by –1 and add.
–2x3
+ 5x2
– x + 8
–2x3
+ 3x – 4– Add the opposite
No change –2x3
+ 5x2
– x + 8
2x3
– 3x + 4+

(–2x3
+ 5x2
– x + 8) – (–2x2
+ 3x – 4)
Use a vertical format. To subtract, you add the opposite. This means you
multiply each term in the subtracted polynomial by –1 and add.
–2x3
+ 5x2
– x + 8
–2x3
+ 3x – 4–
5x2
– 4x + 12
–2x3
+ 5x2
– x + 8
2x3
– 3x + 4+
SOLUTION

(3x2
– 5x + 3) – (2x2
– x – 4)
SOLUTION
Use a horizontal format.
(3x2
– 5x + 3) – (2x2
– x – 4) = (3x2
– 5x + 3) + (–1)(2x2
– x – 4)
= x2
– 4x + 7
= (3x2
– 5x + 3) – 2x2
+ x + 4
= (3x2
– 2x2
) + (– 5x + x) + (3 + 4)

Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
f(x) = x
2
– 3x
4
– 71
2
SOLUTION
The function is a polynomial function.
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.
Its standard form is f(x) = –3x
4
+ x
2
– 7.
1
2

The function is not a polynomial function because the
term 3
x
does not have a variable base and an exponent
that is a whole number.
SOLUTION
f(x) = x
3
+ 3
x

SOLUTION
f(x) = 6x
2
+ 2x
–1
+ x
The function is not a polynomial function because the term
2x–1
has an exponent that is not a whole number.

SOLUTION
The function is a polynomial function.
It has degree 2, so it is a quadratic function.
The leading coefficient is π.
Its standard form is f(x) = πx
2
– 0.5x – 2.
f(x) = – 0.5x + πx
2
– 2 2

EVALUATING A POLYNOMIAL FUNCTION
Find f(-2) if f(x) = 3x2
– 2x – 6
f(-2) = 3(-2)2
– 2(-2) – 6
f(-2) = 12 + 4 – 6
f(-2) = 10

Find f(2a) if f(x) = 3x2
– 2x – 6
f(2a) = 3(2a)2
– 2(2a) – 6
f(2a) = 12a2
– 4a – 6

Find f(m + 2) if f(x) = 3x2
– 2x – 6
f(m + 2) = 3(m + 2)2
– 2(m + 2) – 6
f(m + 2) = 3(m2
+ 4m + 4) – 2(m + 2) –
6
f(m + 2) = 3m2
+ 12m + 12 – 2m – 4 – 6
f(m + 2) = 3m2
+ 10m + 2

Find 2g(-2a) if g(x) = 3x2
– 2x – 6
2g(-2a) = 2[3(-2a)2
– 2(-2a) – 6]
2g(-2a) = 2[12a2
+ 4a – 6]
2g(-2a) = 24a2
+ 8a – 12

POLYNOMIALS

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