UP Board Solutions for Class 10 Maths Chapter 2 Polynomials

 POLYNOMIALS

1. An algebraic expression p(x) of the form p(x) = a₀ , + a₁x+a₂ x²+ … + a_nxⁿ, where a₀, a₁, a₂, …, a_n are real numbers and all the indices of x are non-negative integers, is called a polynomial in x and the highest index n is called the degree of the polynomial, if a_n  0 ,a₀ , + a₁x+a₂ x²+ … + a_nxⁿare called the terms of the polynomial and a₀, a₁, a₂, …, a_n an are called various coefficients of the polynomial f(x). A polynomial in x is said to be in standard form when the terms are written either in increasing order or decreasing order of the indices of x in various terms.

2. Different types of polynomials:

There are four polynomials based on degrees.

(i) Linear polynomial:

A polynomial of degree one is called a linear polynomial. It is of the form ax + b, where a, b € R and a  0.

(ii) Quadratic polynomial :

A polynomial of degree two is called a quadratic polynomial. It is of the form ax² + bx + c, where a, b, c € R and a  0.

(iii) Cubic polynomial:

A polynomial of degree three is called a cubic polynomial. It is of the form ax³ + bx²+ cx + d, where a, b, c, d € R and a 0.

(iv) Biquadratic (or Quartic) polynomial:

A polynomial of degree four is called a biquadratic (or quartic) polynomial. It is of the form ax⁴ + bx²+ cx² + dx + e = 0, where a, b, c, d, e € Rand a  0.

3. Value of the polynomial :

If p(x) is a polynomial in x, and if a is any real constant, then the real number obtained by replacing x by a in p(x), is called the value of p(x) at a, and is denoted by p(a).

4.Zero of a polynomial:

A real number a is called a zero of a polynomial p(x), if p(a) =0.i.e., a zero of a polynomial is the value of the variable for which the value of the polynomial becomes zero.

5. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x)intersects the x-axis.

6. A polynomial of degree n can have at the most n zeroes. So, a quadratic polynomial can have at the most 2 zeroes and a cubic polynomial can have at the most 3 zeroes.

7. If a and are the zeroes of a quadratic polynomial ax² + bx + c, then

a + =  , a = .         

8. If a ,  , y are the zeroes of a cubic polynomial ax³+ bx²+ cx + d  = 0 then

9. The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such thatp(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) <degree q(x).