The degree of a polynomial is determined by the highest power of the variable (or variables) present in the polynomial expression.
For example:
-
In the polynomial
3π₯2+5π₯β1
3
x
2
+
5
x
β
1
, the highest power of the variable
π₯
x
is
2
2
, so the degree of the polynomial is
2
2
.
-
In the polynomial
2π¦3βπ¦+4
2
y
3
β
y
+
4
, the highest power of the variable
π¦
y
is
3
3
, so the degree of the polynomial is
3
3
.
-
In the polynomial
4π₯4π¦2β3π₯π¦+7
4
x
4
y
2
β
3
x
y
+
7
, the highest combined power of the variables
π₯
x
and
π¦
y
is
6
6
(since
π₯
x
has a power of
4
4
and
π¦
y
has a power of
2
2
), so the degree of the polynomial is
6
6
.
The degree of a polynomial helps classify it and understand its behavior when performing mathematical operations like addition, subtraction, multiplication, and division. It's an important concept in algebra and polynomial arithmetic.
Algebraic Identities
Algebraic identities are algebraic equations which are valid for all values. The important algebraic identities used in Class 9 Maths chapter 2 polynomials are listed below:
-
(x + y + z)
2
= x
2
+ y
2
+ z
2
+ 2xy + 2yz + 2zx
-
(x + y)
3
= x
3
+ y
3
+ 3xy(x + y)
-
(x β y)
3
= x
3
β y
3
β 3xy(x β y)
-
x
3
+ y
3
+ z
3
β 3xyz = (x + y + z) (x
2
+ y
2
+ z
2
β xy β yz β zx)
Zeroes of Polynomial
The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero when substituted into it. In other words, if
π(π₯)
P
(
x
)
is a polynomial, then any value
π
a
for which
π(π)=0
P
(
a
)
=
0
is considered a zero (or root) of the polynomial.
For example, consider the polynomial
π(π₯)=π₯2β4
P
(
x
)
=
x
2
β
4
. To find its zeroes, we set
π(π₯)
P
(
x
)
equal to zero and solve for
π₯
x
:
π₯2β4=0
x
2
β
4
=
0
This equation can be factorized as
(π₯β2)(π₯+2)=0
(
x
β
2
)
(
x
+
2
)
=
0
. So, the zeroes of the polynomial are
π₯=2
x
=
2
and
π₯=β2
x
=
β
2
.
In general, a polynomial of degree
π
n
can have at most
π
n
zeroes. These zeroes may be real or complex numbers. The Fundamental Theorem of Algebra states that every polynomial equation of degree
π
n
has exactly
π
n
complex roots (including repeated roots).
The zeroes of a polynomial are important in various mathematical contexts, such as solving equations, graphing functions, and understanding the behavior of polynomial functions.
Remainder Theorem
The Remainder Theorem is a fundamental concept in algebra that relates to polynomial division. It states that if a polynomial
π(π₯)
P
(
x
)
is divided by a linear polynomial of the form
π₯βπ
x
β
a
, then the remainder is equal to
π(π)
P
(
a
)
, where
π
a
is any real number.
In simpler terms, if you divide a polynomial by
π₯βπ
x
β
a
, the remainder you get will be the value of the polynomial evaluated at
π
a
.
For example, let's say we have the polynomial
π(π₯)=π₯2+3π₯β4
P
(
x
)
=
x
2
+
3
x
β
4
and we want to divide it by
π₯β2
x
β
2
. According to the Remainder Theorem, the remainder will be
π(2)
P
(
2
)
, which means we substitute
π₯=2
x
=
2
into the polynomial
π(π₯)
P
(
x
)
. So,
π(2)=(2)2+3(2)β4=4+6β4=6
P
(
2
)
=
(
2
)
2
+
3
(
2
)
β
4
=
4
+
6
β
4
=
6
.
Hence, when
π(π₯)
P
(
x
)
is divided by
π₯β2
x
β
2
, the remainder is
6
6
.
The Remainder Theorem is useful in various mathematical applications, including finding roots of polynomials, evaluating polynomial functions, and proving divisibility properties.
Factorisation of Polynomials
Factorization of polynomials involves expressing a given polynomial as the product of two or more simpler polynomials.
For example, consider the polynomial
π₯2βπ₯β6
x
2
β
x
β
6
. To factorize it, we look for two numbers whose product is
β6
β
6
and whose sum is
β1
β
1
, because the middle term of the polynomial is
π₯
x
and the constant term is
β6
β
6
. These numbers are
β3
β
3
and
2
2
, because
(β3)Γ2=β6
(
β
3
)
Γ
2
=
β
6
and
(β3)+2=β1
(
β
3
)
+
2
=
β
1
. Therefore, we can express
π₯2βπ₯β6
x
2
β
x
β
6
as
(π₯β3)(π₯+2)
(
x
β
3
)
(
x
+
2
)
by using these factors.
This process of factorization helps simplify polynomial expressions and is a fundamental concept in algebra. It allows us to understand the structure of polynomials better and to solve various mathematical problems more efficiently.
Benefits of CBSE Class 9 Maths Notes Chapter 2 Polynomials
-
Concept Clarity
: These notes provide a clear explanation of polynomial concepts, ensuring that students understand the fundamentals thoroughly.
-
Structured Learning
: The notes are organized in a structured manner, covering topics sequentially. This helps students to follow a logical progression in their learning.
-
Comprehensive Coverage
: The notes cover all the essential topics related to polynomials, including definitions, types, factorization, remainder theorem, zeroes, and more. This comprehensive coverage ensures that students have a complete understanding of the chapter.
-
Example Problems
: The notes include solved examples that illustrate how to apply polynomial concepts in different scenarios. These examples help students grasp the application of theory in practical problems.
-
Practice Questions
: Along with solved examples, the notes also provide practice questions at the end of each topic or chapter. These questions allow students to test their understanding and reinforce their learning.
-
Exam Preparation
: By studying these notes, students can effectively prepare for exams. The clear explanations, solved examples, and practice questions help them revise the chapter thoroughly and build confidence for exams.
