# 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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