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 $$r(x) = 0$$

Solution

(i) deg p(x) = deg q(x)We know the formula,Dividend = Divisor x quotient + Remainder$$p(x) = g(x) \times q(x) + r(x)$$So here the degree of quotient will be equal to degree of dividend when the divisor is constant.Let us assume the division of $$4x^2$$ by $$2$$.Here, $$p(x) =$$$$4x^2$$$$g(x) = 2$$$$q(x) =$$ $$2x^2$$ and $$r(x) = 0$$Degree of $$p(x)$$ and $$q(x)$$ is the same i.e., $$2$$.Checking for division algorithm,$$p(x) = g(x) \times q(x) + r(x)$$$$4x^2 = 2(2x^2)$$Hence, the division algorithm is satisfied.(ii) deg q(x) = deg r(x)Let us assume the division of $$x^3 + x$$ by $$x^2$$,Here, p(x) = $$x^3 + x$$, g(x) = $$x^2$$, q(x) = x and r(x) = xDegree of q(x) and r(x) is the same i.e., 1.Checking for division algorithm,$$p(x) = g(x) \times q(x) + r(x)$$$$x^3 + x = x^2 \times x + x$$$$x^3 + x = x^3+ x$$Hence, 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 $$x^4+1$$ by $$x^3$$Here, p(x) = $$x^4+1$$g(x) = $$x^3$$$$q(x) = x$$ and $$r(x) = 1$$Degree of $$r(x)$$ is $$0.$$Checking for division algorithm,$$p(x) = g(x) \times q(x) + r(x)$$$$x^4 + 1 = x^3 \times x + 1$$$$x^4 + 1 = x^4+ 1$$Hence, the division algorithm is satisfied.MathematicsRS AgarwalStandard IX

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