Chapter 2 : Polynomials - Ncert Solutions for Class 10 Maths CBSE

Chapter 2 - Polynomials Excercise 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.

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.

Chapter 2 - Polynomials Excercise 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.

Chapter 2 - Polynomials Excercise Ex. 2.3

Solution 1

Quotient = x - 3 

Remainder = 7x - 9 

Quotient = x² + x - 3

Remainder = 8

Quotient = -x² - 2

Remainder = -5x + 10 

Concept insight: While dividing one polynomial by another, first arrange the polynomial in descending powers of the variable. In the process of division, be careful about the signs of the coefficients of the terms of the polynomials. After performing division, one can check his/her answer obtained by the division algorithm which is as below:

Dividend = Divisor x Quotient + Remainder 

Also, remember that the quotient obtained is a polynomial only.

Solution 2

(i) The polynomial 2t⁴ + 3t³ - 2t² - 9t - 12 can be divided by the polynomial t² - 3 = t² + 0.t - 3 as follows: 

Since the remainder is 0, t² - 3 is a factor of 2t⁴ + 3t³ - 2t² - 9t - 12 . 

(ii) The polynomial 3x⁴ + 5x³ - 7x² + 2x + 2 can be divided by the polynomial x² + 3x + 1 as follows:

Since the remainder is 0, x² + 3x + 1 is a factor of 3x⁴ + 5x³ - 7x² + 2x + 2 

(iii) The polynomial x⁵ - 4x³ + x² + 3x + 1 can be divided by the polynomial x³ - 3x + 1 as follows: 

Since the remainder is not equal to 0, x³ - 3x + 1 is not a factor of x⁵ - 4x³ + x² + 3x + 1. 

Concept insight:  A polynomial g(x) is a factor of another polynomial p(x) if the remainder obtained on dividing p(x) by g(x) is zero and not just a constant. While changing the sign, make sure you do not change the sign of the terms which were not involved in the previous operation. For example in the first step of (iii), do not change the sign of 3x + 1.

Solution 3

Let p(x) = 3x⁴ + 6x³ - 2x² - 10x -5 

Now, x² + 2x + 1 = (x + 1)²

So, the two zeroes of x² + 2x + 1 are -1 and -1.

Concept insight: Remember that if (x - a) and (x - b) are factors of a polynomial, then (x - a)(x - b) will also be a factor of that polynomial. Also, if a is a zero of a polynomial p(x), where degree of p(x) is greater than 1, then (x - a) will be a factor of p(x), that is when p(x) is divided by (x - a), then the remainder obtained will be 0 and the quotient will be a factor of the polynomial p(x). To cross check your answer number of zeroes of the polynomial will be less than or equal to the degree of the polynomial.

Solution 4

Divided, p(x) = x³ - 3x² + x + 2 

Quotient = (x - 2)

Remainder = (-2x + 4)

Let g(x) be the divisor.

According to the division algorithm, 

Dividend = Divisor x Quotient + Remainder

Concept insight: When a polynomial is divided by any other non-zero polynomial, then it satisfies the division algorithm which is as below:

Dividend = Divisor x Quotient + Remainder

Divisor x Quotient = Dividend - Remainder 

So, from this relation, the divisor can be obtained by dividing the result of (Dividend - Remainder) by the quotient.

Solution 5

According to the division algorithm, if p(x) and g(x) are two polynomials with g(x)  0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) x q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

(i)   Degree of quotient will be equal to degree of dividend when divisor is constant.

Let us consider the division of 18x² + 3x + 9 by 3.

Here, p(x) = 18x² + 3x + 9 and g(x) = 3

q(x) = 6x² + x + 3 and r(x) = 0

Here, degree of p(x) and q(x) is the same which is 2.

Checking:

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

Thus, the division algorithm is satisfied.

(ii)   Let us consider the division of 2x⁴ + 2x by 2x³,

Here, p(x) = 2x⁴ + 2x and g(x) = 2x³

q(x) = x and r(x) = 2x

Clearly, the degree of q(x) and r(x) is the same which is 1.

Checking,

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

2x⁴ + 2x = (2x³ ) x x + 2x

2x⁴ + 2x = 2x⁴ + 2x

Thus, the division algorithm is satisfied.

(iii)   Degree of remainder will be 0 when remainder obtained on division is a constant.

Let us consider the division of 10x³ + 3 by 5x².

Here, p(x) = 10x³ + 3 and g(x) = 5x²

q(x) = 2x and r(x) = 3

Clearly, the degree of r(x) is 0.

Checking:

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

10x³ + 3 = (5x² ) x 2x + 3

10x³ + 3 = 10x³ + 3

Thus, the division algorithm is satisfied.

Concept insight: In order to answer such type of questions, one should remember the division algorithm. Also, remember the condition on the remainder polynomial r(x). The polynomial r(x) is either 0 or its degree is strictly less than g(x). The answer may not be unique in all the cases because there can be multiple polynomials which satisfies the given conditions.