Question

Give examples of polynomials p(x), g(x) and r(x) which satisfy the division algorithm and
(i) deg p(x) = deg q(x)​

Answer 1

Answer:

(i) Let us assume the division of 6x2 + 2x + 2 by 2

Here, 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 i.e. 2.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

Or, 6x2 + 2x + 2 = 2x (3x2 + x + 1)

Hence, division algorithm is satisfied.

(ii) 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) Let us assume the division of x3+ 1 by 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.

Hope you understand

Please make it a brainlist answer

Answer 2

$\bf{Solution:-}$

let,

$\bf{p(x)=2x²+5x+2}$ [degree of p(x)=2]

$\bf{q(x)=2x²+2}$ [degree of q(x)=2]

Therefore, g(x)=2

r(x)=0

$\bf\fbox{p(x)=g(x).q(x)+r(x)}$

$\bf{2x²+5x+2=2×(2x²+2)+0}$

$\bf{ = 2x²+4+0}$