Chapter 2 – Polynomials

Exercise 2.1

Question 1:

The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i)

(ii)

(iii)

(iv)

(v)

(v)

Answer:

(i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.

(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.

(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.

(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.

(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

Exercise 2.2

Question 1:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

Answer:

The value of

is zero when x − 4 = 0 or x + 2 = 0, i.e., when x = 4 or x = −2

Therefore, the zeroes of

are 4 and −2.

Sum of zeroes =

Product of zeroes

The value of 4s² − 4s + 1 is zero when 2s − 1 = 0, i.e.,

Therefore, the zeroes of 4s² − 4s + 1 are

and

Sum of zeroes =

Product of zeroes

The value of 6x² − 3 − 7x is zero when 3x + 1 = 0 or 2x − 3 = 0, i.e.,

Therefore, the zeroes of 6x² − 3 − 7x are

Sum of zeroes =

Product of zeroes =

The value of 4u² + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = −2

Therefore, the zeroes of 4u² + 8u are 0 and −2.

Sum of zeroes =

Product of zeroes =

The value of t² − 15 is zero when

, i.e., when

Therefore, the zeroes of t² − 15 are

and

Sum of zeroes =

Product of zeroes =

The value of 3x² − x − 4 is zero when 3x − 4 = 0 or x + 1 = 0, i.e., when

or x = −1

Therefore, the zeroes of 3x² − x − 4 are

and −1.

Sum of zeroes =

Product of zeroes

Question 2:

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

Answer:

Let the polynomial be

, and its zeroes be

and

Therefore, the quadratic polynomial is 4x² − x − 4.

Let the polynomial be

, and its zeroes be

and

Therefore, the quadratic polynomial is 3x² −

x + 1.

Let the polynomial be

, and its zeroes be

and

Therefore, the quadratic polynomial is

Let the polynomial be

, and its zeroes be

and

Therefore, the quadratic polynomial is

Let the polynomial be

, and its zeroes be

and

Therefore, the quadratic polynomial is

Let the polynomial be

Therefore, the quadratic polynomial is

Exercise 2.3

Question 1:

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:

(i)

(ii)

(iii)

Answer:

Quotient = x − 3

Remainder = 7x − 9

Quotient = x² + x − 3

Remainder = 8

Quotient = −x² − 2

Remainder = −5x +10

Question 2:

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

Answer:

Since the remainder is 0,

Hence,

is a factor of

Since the remainder is 0,

Hence,

is a factor of

Since the remainder

Hence,

is not a factor of

Question 3:

Obtain all other zeroes of

, if two of its zeroes are

Answer:

Since the two zeroes are

is a factor of

Therefore, we divide the given polynomial by

We factorize

Therefore, its zero is given by x + 1 = 0

x = −1

As it has the term

, therefore, there will be 2 zeroes at x = −1.

Hence, the zeroes of the given polynomial are

, −1 and −1.

Question 4:

On dividing

by a polynomial g(x), the quotient and remainder were x − 2 and − 2x + 4, respectively. Find g(x).

Answer:

g(x) = ? (Divisor)

Quotient = (x − 2)

Remainder = (− 2x + 4)

Dividend = Divisor × Quotient + Remainder

g(x) is the quotient when we divide

Question 5:

Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) deg r(x) = 0

Answer:

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) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x)

Degree of a polynomial is the highest power of the variable in the polynomial.

(i) deg p(x) = deg q(x)

Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).

Let us assume the division of

by 2.

Here, p(x) =

g(x) = 2

q(x) =

and r(x) = 0

Degree of p(x) and q(x) is the same i.e., 2.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

= 2(

)

Thus, the division algorithm is satisfied.

(ii) deg q(x) = deg r(x)

Let us assume the division of x³ + x by x²,

Here, p(x) = x³ + x

g(x) = x²

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

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

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x³ + x = (x²) × x + x

x³ + x = x³ + x

Thus, the division algorithm is satisfied.

(iii)deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of x³ + 1by x².

Here, p(x) = x³ + 1

g(x) = x²

q(x) = x and r(x) = 1

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

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x³ + 1 = (x²) × x + 1

x³ + 1 = x³ + 1

Thus, the division algorithm is satisfied.

Exercise 2.4

Question 1:

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

Answer:

(i)

Therefore,

, 1, and −2 are the zeroes of the given polynomial.

Comparing the given polynomial with

, we obtain a = 2, b = 1, c = −5, d = 2

Therefore, the relationship between the zeroes and the coefficients is verified.

(ii)

Therefore, 2, 1, 1 are the zeroes of the given polynomial.

Comparing the given polynomial with

, we obtain a = 1, b = −4, c = 5, d = −2.

Verification of the relationship between zeroes and coefficient of the given polynomial

Multiplication of zeroes taking two at a time = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5

Multiplication of zeroes = 2 × 1 × 1 = 2

Hence, the relationship between the zeroes and the coefficients is verified.

Question 2:

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively.