from a handpicked tutor in LIVE 1-to-1 classes

# 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) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0

**Solution:**

The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x),

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

The degree of a polynomial is the highest power of the variable in the polynomial.

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

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

Let us assume the division of 6x^{2} + 2x + 2 by 2

p(x) = 6x^{2} + 2x + 2

g(x) = 2

q(x) = 3x^{2} + x + 1

r(x) = 0

Degree of p(x) and q(x) is same that is 2.

Checking for division algorithm:

p(x) = g(x) × q(x) + r(x)

= 2(3x^{2} + x + 1) + 0

= 6x^{2} + 2x + 2

Thus, the division algorithm is satisfied.

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

Let us assume the division of x^{3} + x by x^{2}

p(x) = x^{3} + x

g(x) = x^{2}

q(x) = x

r(x) = x

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

Checking for division algorithm

p(x) = g(x) × q(x) + r(x)

= (x^{2} × x ) + x

= x^{3} + x

Thus, the division algorithm is satisfied.

(iii) deg r(x) = 0

The degree of the remainder will be 0 when the remainder turns out to be a constant.

Let us assume the division of x^{3} + 1 by x^{2}.

p(x) = x^{3} + 1

g(x) = x^{2}

q (x) = x

r(x) = 1

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

Checking for division algorithm

p(x) = g (x) × q (x) + r (x)

= (x^{2} × x) + 1

= x^{3} + 1

Thus, the division algorithm is satisfied.

**☛ Check: **Class 10 Maths NCERT Solutions Chapter 2

**Video Solution:**

## 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) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0

NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.3 Question 5

**Summary:**

Examples of polynomials p(x), g(x), q(x) and r(x) which satisfy the division algorithm where (i) deg p(x) = deg q(x) is p(x) = 6x^{2} + 2x + 2, g(x) = 2, q(x) = 3x^{2} + x + 1 and r(x) = 0 (ii) deg q(x) = deg r(x) is p(x) = x^{3} + x, g(x) = x^{2}, q(x) = x and r(x) = x (iii) deg r(x) = 0 is p(x) = x^{3} + 1, g(x) = x^{2}, q (x) = x and r(x) = 1 respectively.

**☛ Related Questions:**

- 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(ii) p(x) = x4 - 3x2 + 4x + 5, g(x) = x2 + 1 - x(iii) p(x) = x4 - 5x + 6, g(x) = 2 - x2
- 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(ii) x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2(iii) x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1
- Obtain all other zeroes of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeroes are √ 5/3 and - √ 5/3
- 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).

visual curriculum