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 6x² + 2x + 2 by 2

p(x) = 6x² + 2x + 2

g(x) = 2

q(x) = 3x² + 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(3x² + x + 1) + 0

= 6x² + 2x + 2

Thus, the division algorithm is satisfied.

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

Let us assume the division of x³ + x by x²

p(x) = x³ + x

g(x) = x²

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)

= (x² × x ) + x

= x³ + 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 x³ + 1 by x².

p(x) = x³ + 1

g(x) = x²

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)

= (x² × x) + 1

= x³ + 1

Thus, the division algorithm is satisfied.

☛ Check: Class 10 Maths NCERT Solutions Chapter 2

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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) = 6x² + 2x + 2, g(x) = 2, q(x) = 3x² + x + 1 and r(x) = 0 (ii) deg q(x) = deg r(x) is p(x) = x³ + x, g(x) = x², q(x) = x and r(x) = x  (iii) deg r(x) = 0 is p(x) = x³ + 1, g(x) = x², q (x) = x and r(x) = 1 respectively.

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