Question

13. If the polynomial x19 + x17 +x13 +x11 + x7 + x5 + x3 is divided by (x2 + 1), then the remainder is
A) 1
B) x2 +4
C) -x
D) x​

Answer 1

$\textbf{Given polynomial is}$

$x^{19}+x^{17}+x^{13}+x^{11}+x^7+x^5+x^3$

$\textbf{To find:}$

$\text{The remainder when x^{19}+x^{17}+x^{13}+x^{11}+x^7+x^5+x^3 is divided by x^2+1 }$

$\textbf{Solution:}$

$\text{We find the remainder without using actual long division}$

$\text{Consider,}$

$x^{19}+x^{17}+x^13+x^{11}+x^7+x^5+x^3$

$\text{We rewrite the given polynomial in terms of x^2+1 }$

$=x^{17}(x^2+1)+x^{11}(x^2+1)+x^5(x^2+1)+x^3$

$\text{In the last term, add and subtract x }$

$=x^{17}(x^2+1)+x^{11}(x^2+1)+x^5(x^2+1)+x^3+x-x$

$=x^{17}(x^2+1)+x^{11}(x^2+1)+x^5(x^2+1)+x(x^2+1)-x$

$=(x^2+1)(x^{17}+x^{11}+x^5+x)-x$

$\implies\,x^{19}+x^{17}+x^13+x^{11}+x^7+x^5+x^3=(x^2+1)(x^{17}+x^{11}+x^5+x)-x$

$\therefore\textbf{The remainder when \bf\,x^{19}+x^{17}+x^{13}+x^{11}+x^7+x^5+x^3 is \bf\,-x }$

$\textbf{Hence option (C) is correct}$

Find more:

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Answer 2

Answer:

-x² is the answer

Step-by-step explanation:

The step by step explanation is given in the paper attached to the answer. Thank you....

Aarohi.