Question

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

Hint: Polynomials: - A polynomial is an expression consisting of variable and efficiency that involves only the operations of addition, subtraction, multiplication, etc.
General forms of a polynomial $= a{x^2} + bx + c$
A polynomial is denoted as $p(x)\,or\,g(x)$
A polynomial is always divided by another polynomial but with a lesser degree. By degree we mean the higher power of the variable in the respective polynomial i.e if $p(x)\,$ is divided by some $g(x)$ through long division then
Degree of $g(x)$Therefore,

Given ${x^3} + 3{x^2} + 3x + 1$ is divided by $x$ i.e. let $p(x) = {x^3} + 3{x^2} + 3x + 1.......(1)$ and $g(x) = x$
As we see that degree of $g(x)$i.e. I and degree of $p(x)$ i.e. 3
$\Rightarrow$degree of $g(x) <$degree of $p(x)$
Step1:- Make sure the polynomial is written is decreasing order. If any term is missing, use a zero to fill its place.
We have $p(x) = {x^3} + 3{x^2} + 3x + 1\,and\,g(x) = x$
Applying long division
$x\sqrt {{x^3} + 3{x^2} + 3x + 1} ({x^2} + 3x + 3)$

Step 2:- Dividing the $g(x)$ with the highest power i.e $x$ by ${x^3}$ we get ${x^2}$ we known $x:{x^2} = {x^3}$
Step 3:- Subtraitingit and bringing down the other terms.
Step 4: - Now to get $3{x^2},x$ must be multiplied with $3x$. Hence by subtracting it we are left only with $3x + 1$
Step 5: - To get $3x,x$ must be multiplied by 3, and by subtracting it we are left with 1 only.
Step 6: - We see that now, 1 is a constant and Hence out the remainder.

Therefore, we get 1 as a remainder on dividing ${x^3} + 3{x^2} + 3x + 1$by $x$.

Note: we can solve this question with the help of remainder theorem as well when degree of polynomial $g(x) <$degree of $p(x)$
Remainder theorem:
If a polynomial $p(x)$ is divided by $x - a,$ the remainder is $p(a)$
Here as ${x^3} + 3{x^2} + 3x + 1$is divided by $x$ or $x + o$
$\Rightarrow put\,x = 0$in (1)
$\Rightarrow {(0)^3} + 3{(0)^2} + 3(0) + 1$
$\begin{gathered} \Rightarrow 0 + 0 + 0 + 1 \\ \Rightarrow 1 \\ \end{gathered}$
Hence the remainder is 1