# Class 10 NCERT Solutions Maths Chapter 2 - Polynomials

Ex. 2.1

Ex. 2.2

## Polynomials Exercise Ex. 2.1

### Solution 1

(i) The graph of P(x) does not cut the x-axis at all . So, the number of zeroes is 0.

(ii) The graph of P(x) intersects the x-axis at only 1 point.

So, the number of zeroes is 1.

(iii) The graph of P(x) intersects the x-axis at 3 points.

So, the number of zeroes is 3.

(iv) The graph of P(x) intersects the x-axis at 2 points.

So, the number of zeroes is 2.

(v) The graph of P(x) intersects the x-axis at 4 points.

So, the number of zeroes is 4.

(vi) The graph of P(x) intersects the x-axis at 3 points.

So, the number of zeroes is 3.

At all these points where the graph intersects x axis the value of the polynomial y = p(x) will be zero.

(ii) The graph of P(x) intersects the x-axis at only 1 point.

So, the number of zeroes is 1.

(iii) The graph of P(x) intersects the x-axis at 3 points.

So, the number of zeroes is 3.

(iv) The graph of P(x) intersects the x-axis at 2 points.

So, the number of zeroes is 2.

(v) The graph of P(x) intersects the x-axis at 4 points.

So, the number of zeroes is 4.

(vi) The graph of P(x) intersects the x-axis at 3 points.

So, the number of zeroes is 3.

**Concept insight:**Since the polynomial p(x) given here is a polynomial in variable x, so to find the number of zeroes, we look at the number of points where the graph intersects or touches the x-axis and not the y-axis.At all these points where the graph intersects x axis the value of the polynomial y = p(x) will be zero.

## Polynomials Exercise Ex. 2.2

### Solution 1

So, the zeroes of x² - 2x - 8 are 4 and -2.

**Concept insight:**The zero of a polynomial is that value of the variable which when substituted in the polynomial makes its value 0.

When a quadratic polynomial is equated to 0, then the values of the variable obtained are the zeroes of that polynomial. The relationship between the zeroes of a quadratic polynomial with its coefficients is very important. Also, while verifying the above relationships, be careful about the signs of the coefficients.

### Solution 2

(i) Let the required polynomial be ax² + bx + c, and let its zeroes and

If a = 4k, then b = -k, c = -4k

Therefore, the quadratic polynomial is k(4x

^{ 2}- x - 4), where k is a real number .One quadratic polynomial which fits the given condition is 4x

(ii) Let the polynomial be ax² + bx + c, and let its zeroes be and

^{ 2}- x - 4, when k = 1.(ii) Let the polynomial be ax² + bx + c, and let its zeroes be and

One quadratic polynomial which fits the given condition is 3x

^{2}- 3x + 1, when k = 1.(iii) Let the polynomial be ax² + bx + c, and let its zeroes be and

One quadratic polynomial which fits the given condition is x

^{2}+ x, when k = 1.(iv) Let the polynomial be ax² + bx + c, and let its zeroes be and

Therefore, the quadratic polynomial is k(x² - x + 1),where k is a real number .

One quadratic polynomial which fits the given condition is x² - x + 1, when k = 1.

(v) Let the polynomial be ax² + bx + c, and its zeroes be and

(v) Let the polynomial be ax² + bx + c, and its zeroes be and

Therefore, the quadratic polynomial is k(4x² + x + 1),where k is a real number .

One quadratic polynomial which fits the given condition is 4x² + x + 1,when k = 1.

(vi) Let the polynomial be ax² + bx + c.

(vi) Let the polynomial be ax² + bx + c.

Therefore, the quadratic polynomial is k(x² - 4x + 1),where k is a real number .

One quadratic polynomial which fits the given condition is x² - 4x + 1,when k = 1.

Alternatively If the sum and the product of the zeroes of a quadratic polynomial is given then polynomial is given by x

**Concept insight:**Since the sum and product of zeroes gives 2 relations between three unknowns so we assign a value to the variable a and obtain other values.Alternatively If the sum and the product of the zeroes of a quadratic polynomial is given then polynomial is given by x

^{2 }- (sum of the roots)x + product of the roots, where k is a constant. And the simplest polynomial will be the one in which k = 1.