Find the remainder when $f(x)$ divided by $(x^2 + x + 1)(x+1)$.

Question

When a polynomial $f(x)$ is divided by $x^2 + x + 1$ and $(x+1)^2$, the remainder are $x+5$ and $x-1$ respectively. Find the remainder when $f(x)$ is divided by $(x^2 + x + 1)(x+1)$.

First, I let the remainder be $Ax^2 + Bx +C$, then I try to find the values of $A$, $B$ and $C$. There is $3$ unknowns so we need three equations but I can only get two equations.

Answer 1

Hint $\ f = x\!+\!5 + \color{#c00}c\,(\overbrace{x^2\!+\!x\!+\!1}^{\large g})\ $ and $\ {-}2 = \color{#0a0}{f\bmod x\!+\!1} = f(-1) = 4\!+\!c\iff \color{#c00}{c = -6}$

See this Remark for more detail, and the relationship to CRT and Lagrange interpolation.

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Remark $\:\!\ $ Or use $\,\ fg\bmod hg\, =\, [f\bmod h]\,g\,= $ mod Distributive Law $ $ algorithmically

$$\,\ \ \begin{align} f-(x\!+\!5)\ \bmod\ (x\!+\!1)g\, =\ &\left[\dfrac{f-(x\!+\!5)}{g}\bmod\, x\!+\!1\right]g\ \ \ {\rm by}\ \ \ f\equiv x\!+\!5\!\!\!\pmod{\!g}\\[.3em] =\ &\left[\dfrac{f(-1)-4}{1}\right]g\ \ \ \,{\rm by}\ \ \ g\bmod x\!+\!1 = g(-1) = 1\\[.3em] =\ &[\color{#c00}{\,-6}\,]\,g\ \ \ {\rm by}\ \ f(-1)=\color{#0a0}{f\bmod x\!+\!1} = x\!-\!1\bmod x\!+\!1 = -2 \end{align}$$

Note that we did not need to solve any equations - only evaluate polynomials at $\,x=-1$
The same method works generally:

Lemma $\,\ \bbox[7px,border:1px solid #c00]{f\bmod (x\!-\!a)g\, =\, f_g + \left[\dfrac{f(a)- f_g(a)}{g(a)}\right]g}\ \ $ if $\ g(a)\neq 0,\,$ for $\,f_g := f\bmod g$

$\begin{align}{\bf Proof}\ \ \ \ f-f_g\ \bmod\ (x\!-\!a)g\, =\ &\left[\dfrac{f-f_g}{g}\bmod\, x\!-\!a\right]g\\[.3em] =\ &\left[\dfrac{f(a)-f_g(a)}{g(a)}\right]g \end{align}$

The formula does not require solving equations, only evaluating a few polynomials at $\,x=a$

Note $ $ The $\!\bmod\!$ Distributive Law can be viewed as an equivalent "shifty" operational reformulation of CRT = Chinese Remainder Theorem, as I explain in the end of my Remark here. It is often more convenient to apply in practice because of its operational nature, e.g. here are many examples.

Answer 2

We know that $$f(x)=Q_1(x)(x^2 + x + 1)+x+5=Q_2(x)(x+1)^2+x-1$$ for some polynomials $Q_1$ and $Q_2$. Moreover, for some polynomial $Q_3$, $$f(x)=Q_3(x)(x^2 + x + 1)(x+1)+Ax^2+Bx+C.$$ Hence $$f(-1)=-1-1=A(-1)^2+B(-1)+C.$$ Now consider the complex zeros $w_1$ and $w_2$ of $x^2 + x + 1$. Then $$f(w_1)=w_1+5=Aw_1^2+Bw_1+C\quad,\quad f(w_2)=w_2+5=Aw_2^2+Bw_2+C.$$ By solving the linear system of the three equations we find that $A = -6$, $B = -5$, $C = -1$.

Answer 3

$$p(x)\equiv (x-1)\pmod{(x+1)^2} $$ implies $p(x)\equiv (x-1) \equiv -2 \pmod{(x+1)}$. We know that $p(x)\equiv(x+5)\pmod{x^2+x+1}$, or $$ p(x) = (x+5) + (x^2+x+1) r(x) $$ hence $$ p(x) \equiv 4 + r(x)\pmod{(x+1)}$$ and $r(x)\equiv -6\pmod{(x+1)}$. It follows that $$ p(x) = (x+5)+(x^2+x+1)(-6+(x+1)s(x)) $$ and $$ p(x)\equiv \color{red}{(x+5)-6(x^2+x+1)} \pmod{(x+1)(x^2+x+1)}.$$

Answer 4

The remainder when dividing by $(x+1)^2$ was $x-1$, so the remainder when dividing by just $x+1$ must be $-2$.

We know that $f(x)$ is of the form $$ f(x) = q(x)(x^2 + x + 1)(x+1) + Ax^2 + Bx + C $$ where $q(x)$ is a polynomial and $Ax^2 + Bx + C$ has the same remainders as $f$ does when divided by $x+1$ and by $x^2 + x + 1$. Now we just calculate: $$ \frac{Ax^2 + Bx + C}{x+1} = Ax + B-A+\frac{C-B+A}{x+1} $$ so we must have $C-B+A = -2$. That's one equation down. Now we deal with $x^2 + x + 1$: $$ \frac{Ax^2 + Bx + C}{x^2 + x + 1} = Ax + \frac{(B-A)x + C-A}{x^2 + x + 1} $$ so we know that $(B-A)x + C-A = x+5$. As I alluded to in the comments to the question above, this is actually two equations in one, as this is an equality of functions. The slopes must be equal, and the constant terms must be equal. This gives you the remaining two equations $B-A = 1$ and $C-A = 5$.

Answer 5

Using the Generalized Euclidean Algorithm, we get $$ (x+2)\left(x^2+x+1\right)-(x+1)(x+1)^2=1\tag1 $$ Therefore, we have $$ \begin{align} (x+2)\left(x^2+x+1\right)&\equiv1&&\pmod{(x+1)^2}\\ (x+2)\left(x^2+x+1\right)&\equiv0&&\pmod{x^2+x+1} \end{align}\tag2 $$ and $$ \begin{align} -(x+1)(x+1)^2&\equiv0&&\pmod{(x+1)^2}\\ -(x+1)(x+1)^2&\equiv1&&\pmod{x^2+x+1} \end{align}\tag3 $$ Thus, to get $$ \begin{align} f(x)&\equiv x-1&&\pmod{(x+1)^2}\\ f(x)&\equiv x+5&&\pmod{x^2+x+1} \end{align}\tag4 $$ we can use $x-1$ times $(2)$ plus $x+5$ times $(3)$ to get $\bmod{(x+1)^2\left(x^2+2x+1\right)}$ $$ \begin{align} f(x) &\equiv(x-1)(x+2)\left(x^2+x+1\right)-(x+5)(x+1)(x+1)^2\\ &=-6x^3-18x^2-17x-7\\ &\equiv-6x^2-5x-1\pmod{(x+1)\left(x^2+2x+1\right)}\tag5 \end{align} $$

Answer 6

Write $$ f(x)=k(x)(x^2+x+1)+x+5$$

and $$ f(x)=q(x)(x+1)^2+x-1 \Longrightarrow f(-1) = -2$$

So $$-2 = f(-1) = k(-1)\cdot 1 +4 \Longrightarrow k(-1) = -6$$

thus

$$ k(x) = p(x)(x+1)-6 $$ and finally \begin{eqnarray} f(x) &= &(p(x)(x+1)-6)(x^2+x+1)+x+5 \\ &=& p(x)(x+1)(x^2+x+1)-6x^2-5x-1 \end{eqnarray}