# NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

## NCERT Solutions for Class 10 Maths Chapter 2 Polynomials - Free PDF

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Q1. What is the difference between remainder theorem and factor theorem?

Remainder theorem states that if a polynomial p(x) is divided by (x - a), then the remainder is obtained by evaluating the expression p(a).

While, Factor theorem states that (x - a) will be a factor of polynomial p(x) only if the remainder obtained by evaluating the expression p(a) equals zero i.e; p(a) = 0.

For example: let p(x) = x² - 6x + 9, then find the remainder when it is divided by (x - 1).

So, according to the remainder theorem, remainder is obtained by evaluating the expression p(1)

p(1) = 1² - 6(1) + 9

= 1 - 6 + 9 = 4.

Therefore, the remainder is 4 when p(x) = x² - 6x + 9 is divided by (x - 1).

Now, check whether (x - 3) is the factor of p(x) = x² - 6x + 9.

So, according to factor theorem (x - 3) is the factor of p(x) = x² - 6x + 9 only if p(3) = 0.

p(3) = 3² - 6(3) + 9

= 9 - 18 + 9 = 0.

Therefore, (x - 3) is the factor of p(x) = x² - 6x + 9.

Q2: What is the geometrical meaning of the zeroes of a Polynomial?

For a linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point at (-b/a , 0). Therefore, the geometrical meaning of the zeroes of linear polynomial ax + b, a ≠ 0 is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.

For a quadratic polynomial ax² + bx + c, a ≠ 0, the graph of the equation y = ax² + bx + c is parabola, which is ‘U’ shaped either open upwards or open downwards depending on whether a > 0 or a < 0. Therefore, the geometrical meaning of a quadratic polynomial is that it can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This means that a polynomial of degree 2 has at most two zeroes.

In general, for any given polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, the geometrical meaning for a polynomial p(x) of degree n is that it has at most n zeroes.

Q3. What are the relationships between zeroes and coefficients of (a) Quadratic polynomial (b) Cubic polynomial?

For quadratic polynomial ax² + bx + c, a ≠ 0, if α and β are its zeroes, then relationship between the zeroes and coefficients of quadratic polynomial is:

Sum of zeroes = α + β = -b/a = -(coefficient of x) / coefficient of x²

Product of zeroes = αβ = c/a = constant term / coefficient of x².

For cubic polynomial ax³ + bx² + cx + d, a ≠ 0, if α, β and γ are its zeroes, then relationship between the zeroes and coefficients of cubic polynomial is:

Sum of zeroes = α + β + γ = -b/a = -(coefficient of x²) / coefficient of x³.

Sum of product of zeroes taken two at a time = αβ + βγ + γα = c/a = coefficient of x / coefficient of x³.

Product of zeroes = α β γ = d/a = constant term / coefficient of x³.