Polynomial Class 10 Notes: Chapter 2

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.

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’.
– 8xy2, 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: 2x2y+6xy+3x+9 which is the sum of four terms: 2x2y, 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.

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.

8x3+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 3x2+x+5 is 2, as the highest power of x in the given expression is x2.

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:

1. a) Monomial – A polynomial having one term. Example: 5x, 2x2, 3xy
2. b) Binomial – A polynomial having two terms. Example: 6x2+2x, 3x+7
3. a) Trinomial – A polynomial having three terms. Example: 2x2+6x+9

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.

A quadratic polynomial is a polynomial having two degrees.

For example, 5x2+3x+6 is a quadratic polynomial.

Cubic Polynomial

A cubic polynomial is a polynomial having three degrees.
For example, 2x3+5x2+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.

1. A linear polynomial consists of only one zero,
2. A quadratic polynomial can have at most two zeros.
3. A cubic polynomial can have at most 3 zeros.

Factorization of Polynomials

By separating the middle term, quadratic polynomials can be factorized.

For example, have a look on the polynomial 6x2+17x+5

Splitting the middle term:

As we can see, 17x is the middle term in the polynomial 6x2+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 x2 and the constant term)

17 can be expressed as (15) +(2), as 15×2=30

Thus, 6x2+17x+5=6x2+15x+2x+5

Now, we will identify the common factors in individual groups

6x2+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)

If α and β are the roots of a quadratic polynomial ax2+bx+c, then,

α + β = -b/a

Sum of zeroes = -coefficient of x /coefficient of x2

αβ = c/a

Product of zeroes = constant term / coefficient of x2

For Cubic Polynomial

If α,β and γ are the roots of a cubic polynomial ax3+bx2+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=a2+2ab+b2

2. (a−b)2=a2−2ab+b2

3. (x+a)(x+b)=x2+(a+b)x+ab

4. a2−b2=(a+b)(a−b)

5. a3−b3=(a−b)(a2+ab+b2)

6. a3+b3=(a+b)(a2−ab+b2)

7. (a+b)3=a3+3a2b+3ab2+b3

8. (a−b)3=a3−3a2b+3ab2−b3