Here are the notes for **CBSE** **Class 10 Maths Chapter 2 Polynomia**l. 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’.

– 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**.

**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:

- a) Monomial – A polynomial having one term. Example: 5x, 2x
^{2}, 3xy - b) Binomial – A polynomial having two terms. Example: 6x
^{2}+2x, 3x+7 - a) Trinomial – A polynomial having three terms. Example: 2x
^{2}+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.

**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.

- A linear polynomial consists of only one zero,
- A quadratic polynomial can have at most two zeros.
- 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 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}