# PolynomialsExercise – 2.3

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

(i) p(x) = x3-3x2+5x-3, g(x) = x2-2

Solution:

Quotient = x – 2 and remainder = 7x – 9

(ii) p(x) = x4-3x2+4x+5, g(x) = x2+1-x

Solution:

Quotient = x2 + x - 3 and remainder = 8

(iii) p(x) = x4-5x+6, g(x) = 2 – x2

Solution:

Quotient = -x2 -2 and remainder = -5x + 10

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

(i) t2 – 3, 2t4 + 3t3 – 2t2 – 9t – 12

Solution:

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

(ii) x2 + 3x + 1, 3x4 + 5x3 – 7x2 +2x +2

Solution:

Since, remainder is 0, therefore the polynomial x2 + 3x + 1 is a factor of the polynomial 3x4 + 5x3 – 7x2 +2x +2.

(iii) x3 -3x + 1, x5 -4x3 +x2 +3x +1

Solution:

The polynomial x3 -3x + 1 is not a factor of the polynomial x5 -4x3 +x2 +3x +1 because the remainder is 0 not 2.

3. Obtain all other zeroes of 3x4 +6x3 -2x2 -10x -5, if two of its zeroes are (5/3) and -(5/3).

Solution:

Let p(x) = 3x4 +6x3 -2x2 -10x -5

Therefore, x =√(5/3), x =-√(5/3)

Or

3x2 – 5 = 0

3x2 – 5 is a factor of p(x),

by division method,

p(x) = (3x2 -5) (x2 + 2x +1)

Now, on factoring x2 + 2x +1,

x2 + 2x +1 = 0

x2 + x + x +1 = 0

x (x+1) +1(x+1) = 0

( x+1) (x+1) = 0

x = -1, -1

Hence two other zeroes are -1, -1.

4. On dividing x3 – 3x2 + x +2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x +4, respectively. Find g(x).

Solution:

Given that,

Dividend p(x) = x3 – 3x2 + x +2

Quotient q(x) = x – 2

Remainder r(x) = -2x +4

Divisor g(x) = ?

We know that,

p(x) = g(x)× q(x) + r(x)

x3 – 3x2 + x +2 = g(x)×(x – 2) + (-2x +4)

by division method,

Hence, g(x) = x2 - x +1

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

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

Solution:

The degree of the dividend and the quotient can be equal only if the divisor is a constant (to the degree 0) number.

Let p(x) = 3x2 -6x +5

Let g(x) = 3

Therefore, q(x) = x2 -2x +1 and r(x) = 2

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

Solution:

Let p(x) = 2x2 -4x +3

Let g(x) = x2 -2x +1

Therefore, q(x) = 2 and r(x) = 1

(iii) deg r(x) = 0

Solution:

Let p(x) = 2x2 -4x +3

Let g(x) = x2 -2x +1

Therefore, q(x) = 2 and r(x) = 1