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 $r(x) = 0$

To do:

We have to 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) $p(x), g(x), q(x), r(x)$

deg $p(x) =$ deg $q(x)$

Both $g(x)$ and $r(x)$ are constant terms.

$p(x) = 2x^2+2x + 4$

$g(x) = 2$

$q(x) = x^2 + x + 2$

$r(x) = 0$

(ii) $p(x), g(x), q(x), r(x)$

deg $q(x) =$ deg $r(x)$

This is possible when degrees of both $q(x)$ and $r(x)$ are less than $p(x)$ and $g(x)$.

$p(x) = x^3+ x^2 + x + 1$

$g(x) = x^2 - 1$

$q(x) = x + 1$

$r(x) = x + 2$

(iii) $p(x), g(x), q(x), r(x)$

deg $r(x) = 0$

This is possible when product of $q(x)$ and $g(x)$ form a polynomial whose degree is equal to degree of $p(x)$ and constant term.

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Updated on: 10-Oct-2022

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