Question

# Give examples of polynomials and , which satisfy the division algorithm and (i) deg deg  (ii) deg deg (iii) deg

Medium
Solution
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## (i) deg p(x) = deg q(x) We know the formula, Dividend = Divisor x quotient + Remainder So here the degree of quotient will be equal to degree of dividend when the divisor is constant. Let us assume the division of by . Here, and Degree of and is the same i.e., . Checking for division algorithm, Hence, the division algorithm is satisfied. (ii) deg q(x) = deg r(x) Let us assume the division of by , Here, p(x) = , g(x) = , q(x) = x and r(x) = x Degree of q(x) and r(x) is the same i.e., 1. Checking for division algorithm, Hence, 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 by Here, p(x) = g(x) = and Degree of is Checking for division algorithm, Hence, the division algorithm is satisfied.

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