====== Question 1, Exercise 5.1 ======
Solutions of Question 1 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. 

=====Question 1(i)=====
Find the remainder by using 'Remainder Theorem': $2 x^{3}+3 x^{2}-4 x+1$ is divided by $x+2$.

** Solution. **

Given: $p(x)=2 x^{3}+3 x^{2}-4 x+1$\\
$x-c=x+2 \implies c=-2$.

By Remainder Theorem, we have
\begin{align*}
\text{Remainder} & = p(c) = p(-2) \\
& = 2(-2)^{3}+3 (-2)^{2}-4 (-2)+1 \\
& = -16+12+8+1 \\
&= 5.
\end{align*}
Hence remiander = 5.
=====Question 1(ii)=====
Find the remainder by using 'Remainder Theorem': $x^{4}+2 x^{3}-x^{2}+2 x+3$ is divided by $x-2$.

** Solution. **

Given: \( p(x) = x^{4} + 2x^{3} - x^{2} + 2x + 3 \) \\
\( x - c = x - 2 \implies c = 2 \).

By the Remainder Theorem, we have
\begin{align*}
\text{Remainder} & = p(c) = p(2) \\
& = (2)^{4} + 2(2)^{3} - (2)^{2} + 2(2) + 3 \\
& = 16 + 16 - 4 + 4 + 3 \\
& = 35.
\end{align*}
Hence, the remainder is 35.





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