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

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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,Complete  step by step answer: 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) 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) 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

Find the remainder when \[{x^3} + 3{x^3} + 3x + 1\]is divided by \[x\].

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Find the remainder when \[{x^3} + 3{x^3} + 3x + 1\]is divided by \[x\].

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