Question

1.
Find the remainder when
${x}^{3} + {3x}^{2} + {3x}^{1} + 1$
is divided by
$5 + 2x$
​

Answer 1

Given:

• Dividend = x³ + 3x² +3x + 1
• Divisor = 5 + 2x

Find:

• What is the remainder when x³ + 3x² +3x + 1 is divided by 5 + 2x

Solution:

Here, we apply remainder theorem to obtain the remainder.

Let, 5 + 2x = 0

$\sf \to 2x = - 5$

$\underline{ \boxed{ \sf \color{red} \to x = \frac{ - 5}{2} }}$

So, put value of x in Dividend of x³ + 3x² +3x + 1

$\sf \longrightarrow x³ + 3x² +3x + 1$

$\sf \longrightarrow { \bigg( \dfrac{ - 5}{2} \bigg)}^{3} + 3 { \bigg( \dfrac{ - 5}{2} \bigg)}^{2} +3 \bigg( \dfrac{ - 5}{2} \bigg)+ 1$

$\sf \longrightarrow - 15.625 + 3 (6.25) +3 \bigg( \dfrac{ - 5}{2} \bigg)+ 1$

$\sf \longrightarrow - 15.625 + 18.75 - 7.5+ 1$

$\sf \longrightarrow 3.125 - 6.5$

$\underline{ \boxed{ \color{blue}\sf \longrightarrow - 3.375}}$

Hence, the remainder will be -3.375

Answer 2

Given:

Dividend = x³ + 3x² +3x + 1

Divisor = 5 + 2x

Find:

What is the remainder when x³ + 3x² +3x + 1 is divided by 5 + 2x

Solution:

Here, we apply remainder theorem to obtain the remainder.

Let, 5 + 2x = 0

$\sf \to 2x = - 5$

$\underline{ \boxed{ \sf \color{red} \to x = \frac{ - 5}{2} }}$

So, put value of x in Dividend of x³ + 3x² +3x + 1

$\sf \longrightarrow x³ + 3x² +3x + 1$

$\sf \longrightarrow { \bigg( \dfrac{ - 5}{2} \bigg)}^{3} + 3 { \bigg( \dfrac{ - 5}{2} \bigg)}^{2} +3 \bigg( \dfrac{ - 5}{2} \bigg)+ 1$

$\sf \longrightarrow - 15.625 + 3 (6.25) +3 \bigg( \dfrac{ - 5}{2} \bigg)+ 1$

$\sf \longrightarrow - 15.625 + 18.75 - 7.5+ 1$

$\sf \longrightarrow 3.125 - 6.5$

$\underline{ \boxed{ \color{blue}\sf \longrightarrow - 3.375}}$

Hence, the remainder will be -3.375

hope it helps you ☺☺❤