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Here are the notes for CBSE Class 10 Maths Chapter 2 Polynomial. With several examples, we will cover everything from what is a polynomial and its kinds to algebraic expressions, degree of a polynomial expression, graphical representation of polynomial equations, factorization, and the link between zeroes and the coefficient of a polynomial.
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Algebraic Expressions
Variables and constants, as well as mathematical operators, make up an algebraic expression.
An algebraic expression is a collection of concepts that serve as expression building blocks.
Variables and constants are combined to form a term. In some cases, a term can be an algebraic expression in itself.
Examples of a term – 5 which is just a constant.
– 7x, which is the product of constant ‘7’ and the variable ‘x’
– 2xy, which is the product of the constant ‘2’ and the variables ‘x’ and ‘y’.
– 8xy^{2}, which is the product of 8, x, y and y.
The coefficient is referred to as the constant in each term.
Example of an algebraic expression: 2x^{2}y+6xy+3x+9 which is the sum of four terms: 2x^{2}y, 6xy, 3x and 9.
Any number of terms can be used in an algebraic expression. Each term's coefficient can be any real number. Any number of variables can be found in an algebraic expression. The variables' exponents, on the other hand, must be rational values.
CBSE Class 10 Maths Notes of All Chapters:
CBSE CLASS 10 MATHS NOTES CHAPTER 1 REAL NUMBERS
CBSE CLASS 10 MATHS NOTES CHAPTER 2 POLYNOMIALS
CBSE CLASS 10 MATHS NOTES CHAPTER 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
CBSE CLASS 10 MATHS NOTES CHAPTER 4 QUADRATIC EQUATION
CBSE CLASS 10 MATHS NOTES CHAPTER 5 AIRTHMETIC PROGRESSION
CBSE CLASS 10 MATHS NOTES CHAPTER 6 TRIANGLE
CBSE CLASS 10 MATHS NOTES CHAPTER 7 CONTROL AND COORDINATION
CBSE Class 10 Maths Notes Chapter 8 Introduction To Trigonometry
CBSE CLASS 10 MATHS NOTES CHAPTER 9 SOME APPLICATIONS OF TRIGONOMETRY
CBSE Class 10 Maths Notes Chapter 10 Circle
CBSE CLASS 10 MATHS NOTES CHAPTER 11 CONSTRUCTION
CBSE CLASS 10 MATHS NOTES CHAPTER 12 AREAS RELATED TO CIRCLES
CBSE CLASS 10 MATHS NOTES CHAPTER 13 SURFACE AREA AND VOLUMES
CBSE CLASS 10 MATHS NOTES CHAPTER 14 STATISTICS
CBSE CLASS 10 MATHS NOTES CHAPTER 15 PROBABILITY
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Polynomial
Exponents of rational numbers can be found in algebraic expressions. A polynomial, on the other hand, is an algebraic expression with a whole number as its exponent on any variable.
8x^{3}+2x+5 is an example of a polynomial as well as an algebraic expression .
3x+5√x is not a polynomial as the exponent on x is 1/2 which is not a whole number but it is an example of algebraic expression.
Degree of a Polynomial
The degree of a polynomial in one variable is equal to the largest exponent on the variable in the polynomial.
Example: The degree of the polynomial 3x^{2}+x+5 is 2, as the highest power of x in the given expression is x^{2}.
Types Of Polynomials
Polynomials can be categorised based upon:
a) Number of terms
b) Degree of the polynomial.
Different types of polynomials based upon the number of terms in it:
Types of Polynomials based upon Degree
Linear Polynomial
A linear polynomial is a polynomial having one degree.
For example, 3x+5 is a linear polynomial.
Quadratic Polynomial
A quadratic polynomial is a polynomial having two degrees.
For example, 5x^{2}+3x+6 is a quadratic polynomial.
Cubic Polynomial
A cubic polynomial is a polynomial having three degrees.
For example, 2x^{3}+5x^{2}+9x+15 is a cubic polynomial.
Zeroes of a polynomial
The value of x for which the value of p(x) is 0 is the zero of a polynomial p(x). If k is a p(x) zero, then p(k)=0.
Number of Zeros
Generally, a polynomial having n degrees can have at most n zeros.
Factorization of Polynomials
By separating the middle term, quadratic polynomials can be factorized.
For example, have a look on the polynomial 6x^{2}+17x+5
Splitting the middle term:
As we can see, 17x is the middle term in the polynomial 6x^{2}+17x+5. 17x needs to be expressed as a sum of two terms such that the product of their coefficients is equal to the product of 6 and 5 (coefficient of x^{2} and the constant term)
17 can be expressed as (15) +(2), as 15×2=30
Thus, 6x^{2}+17x+5=6x^{2}+15x+2x+5
Now, we will identify the common factors in individual groups
6x^{2}+2x+15x+5=2x(3x+1)+5(3x+1)
Now we can express it by taking (3x+1) as the common factor:
2x(3x+1) +5(3x+1)=(2x+5)(3x+1)
For Quadratic Polynomial:
If α and β are the roots of a quadratic polynomial ax^{2}+bx+c, then,
α + β = -b/a
Sum of zeroes = -coefficient of x /coefficient of x^{2}
αβ = c/a
Product of zeroes = constant term / coefficient of x^{2}
For Cubic Polynomial
If α,β and γ are the roots of a cubic polynomial ax^{3}+bx^{2}+cx+d, then
α+β+γ = -b/a
αβ +βγ +γα = c/a
αβγ = -d/a
Division Algorithm
Following steps should be followed if we want to divide one polynomial by another.
Step 1: In decreasing order of their degrees, arrange the terms of the dividend and the divisor.
Step 2: Divide the highest degree term of the dividend by the highest degree term of the divisor to get the first term of the quotient. After that, complete the division procedure.
Step 3: The dividend for the next step is the result from the previous division. Repeat this method until the remainder's degree is less than the divisor's degree.
Algebraic Identities
1. (a+b)^{2}=a^{2}+2ab+b^{2}
2. (a−b)^{2}=a^{2}−2ab+b^{2}
3. (x+a)(x+b)=x^{2}+(a+b)x+ab
4. a^{2}−b^{2}=(a+b)(a−b)
5. a^{3}−b^{3}=(a−b)(a^{2}+ab+b^{2})
6. a^{3}+b^{3}=(a+b)(a^{2}−ab+b^{2})
7. (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}
8. (a−b)^{3}=a^{3}−3a^{2}b+3ab^{2}−b^{3}