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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) = x

Degree 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) = 0

Degree 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.

Mathematics
RS Agarwal
Standard IX

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