On this page, you will find Polynomials Class 10 Notes Maths Chapter 2 Pdf free download. CBSE NCERT Class 10 Maths Notes Chapter 2 Polynomials will seemingly help them to revise the important concepts in less time.

## CBSE Class 10 Maths Chapter 2 Notes Polynomials

### Polynomials Class 10 Notes Understanding the Lesson

1. The value of the polynomial p(x) at x = a is p(a).

2. Zeroes of the polynomial p(x) can be find by equating p(x) to zero and solving the equation for

3. If for p(x) = ax^{2} + bx + c = 0, a ≠ 0; α and β are the zeroes, then

4. If for p(x) = ax^{3} + bx^{2} + cx + d = 0; a ≠ 0; α, β,γ are the zeroes, then

α + β + γ = \(\frac{-b}{a}\)

αβ+βγ + αγ = \(\frac{c}{a}\)

αβγ =\(\frac{-d}{a}\)

5. If α and β are the zeroes; then quadratic polynomial will be given by K[x^{2} – Sx + P]

where

S = α +β

P = αβ

K (≠0) is real.

6. The cubic polynomial with zeroes α, β and γ is given by

K[x^{3} – S_{1}x^{2} + S_{1}x^{2} S_{3}]

where

S_{1} = α + β + γ

S_{2 }= αβ + βγ + αγ

S_{3}= αβγ

K(≠ 0) is real.

Degree of a Polynomial:

1. The degree of a polynomial p(x) in x is the highest power of x in p(x)

Note: Expressions like \(\frac{1}{\sqrt{x}}, \frac{1}{x^{2}+1}, \sqrt{x+2}\)

2. (i) Polynomial with degree 1, i.e., polynomial of the form ax + b; a ≠ 0 is called linear polynomial.

(ii) Polynomial with degree 2, i.e., polynomial of the form ax2 + bx + c; a ≠ 0 is called quadratic polynomial.

(iii) Polynomial with degree 3, i.e. polynomial of the form ax^{3} + bx^{2} + cx + d ; a ≠ 0 is called cubic polynomial.

(iv) Polynomial with degree 4, i.e. polynomial of the form ax^{4} + bx^{3} + cx^{2} + dx + e; a ≠ 0 is called biquadratic polynomial.

Geometrical Meaning of the Zeroes of a Polynomial:

1. For any polynomial y = f(x), the number of points on which the graph of y = f(x) intersects at x-axis is called the number of the zeroes of the polynomial and the x-coordinates of these points are called the zeroes of the polynomial y = f(x).

2. Polynomial with degree ‘n’ has maximum ‘n’ number of zeroes. A constant polynomial has no zeroes.

3. Geometrical representation of a linear polynomial is always a straight line.

4. Geometrical representation of a quadratic polynomial is the graph of the shape either open upwards like ‘∪’ or open downwards like ‘∩’ according to a > 0 or a < 0. These curves are called Parabola.

Relationship Between Zeroes and Coefficient of a Quadratic Polynomial:

1. If α and β are the zeroes of the quadratic polynomial p(x) = ax^{2} + bx + c a≠0 then

**Relationship Between Zeroes and Coefficient of a Cubic Polynomial**

1. If α, β and γ are the zeroes of the cubic polynomial

2. A quadratic polynomial p(x) with zeroes α and β is given by

p(x) = K[x^{2} – (α + β)x + αβ]

where K(≠0) is real.

3. A cubic polynomial p(x) with α, β and γ as zeroes is given by

p(x) = K[x^{3} – (α + β + γ)x^{2} + (αβ +βγ + αγ)x – αβγ

where K(≠0) is real.

Division Algorithm for Polynomials:

If p(x) and g(x) are any two polynomials where g(x) ≠ 0. Then on dividing p(x) by g(x), we find other two polynomials q(x) and r(x) such that

p(x) = g(x) x q(x) + r(x);

where deg. of r(x) < deg. of gix)

or Dividend = Divisor x Quotient + Remainder

Note:

- If r(x) = 0, then g(x) will be a factor of p(x) otherwise not.
- If any real number ‘a’ is a zero of the polynomial p(x), then (x – a) will be a factor of p(x).

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