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

Question

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

Filo · Accepted Answer

(i) deg p(x) = deg q(x)We know the formula,Dividend = Divisor x quotient + Remainderp(x)=g(x)×q(x)+r(x)So here the degree of quotient will be equal to degree of dividend when the divisor is constant.Let us assume the division of 4x2 by 2.Here, p(x)=4x2g(x)=2q(x)= 2x2 and r(x)=0Degree of p(x) and q(x) is the same i.e., 2.Checking for division algorithm,p(x)=g(x)×q(x)+r(x)4x2=2(2x2)Hence, the division algorithm is satisfied.(ii) deg q(x) = deg r(x)Let us assume the division of x3+x by x2,Here, p(x) = x3+x, g(x) = x2, q(x) = x and r(x) = xDegree 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+xx3+x=x3+xHence, the division algorithm is satisfied.(iii) deg r(x) = 0Degree of remainder will be 0 when remainder comes to a constant.Let us assume the division of x4+1 by x3Here, p(x) = x4+1g(x) = x3q(x)=x and r(x)=1Degree of r(x) is 0.Checking for division algorithm,p(x)=g(x)×q(x)+r(x)x4+1=x3×x+1x4+1=x4+1Hence, the division algorithm is satisfied.

Question Text	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 $r (x) = 0$
Answer Type	Text solution:1
Upvotes	151

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 $r (x) = 0$

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