Exercises - Developing an Algorithm to Divide Polynomials

1. Perform the following divisions. For each, express your answer in both quotient-remainder form, and then as the sum of a polynomial and proper rational expression.$\require{color}$

1. $(x^3 + 10x^2 + 12x + 4) \div (x+6)$

2. $\cfrac{x^3 - 5x + 4}{x-2}$

3. $(x^3+4x^2-6x+3) \div (x^2-5x+7)$

4. $\cfrac{x^4 - 2x^2 + 1}{3x+1}$

1. $\displaystyle{ \begin{array}[t]{l} {\color{white}x+6 \ )\ x^3 + 10}x^2 + \ 4x - 12\\ x+6 \ \overline{\smash{\raise{.6ex}{) \ }}{\vphantom{)}} x^3 + 10x^2 + 12x + \,4}\\ {\color{white}x+6 \ )\ }\underline{x^3 + \ 6x^2}\\ {\color{white}x+6 \ )\ x^3 + {\ }}4x^2\\ {\color{white}x+6 \ )\ x^3 + {\ }}\underline{4x^2 + \,24x}\\ {\color{white}x+6 \ )\ x^3 + {\ }4x^2} \ -12x\\ {\color{white}x+6 \ )\ x^3 + {\ }4x^2}\underline{\ -12x - 72}\\ {\color{white}x+6 \ )\ x^3 + {\ }4x^2 - 12x + {}} \,76 \end{array} }$

Quotient-remainder form:

$x^2 + 4x - 12$, with remainder $76$

Quotient as a sum of a polynomial and (proper) rational expression:

$\displaystyle{x^2 + 4x - 12 + \cfrac{76}{x+6}}$

2. $\displaystyle{ \begin{array}[t]{l} \hphantom{x-2 \ )\ x^3 + {}}\hphantom{0}x^2 + 2x - 1\\ x-2 \ \overline{\smash{\raise{.6ex}{) \ }}{\vphantom{)}} x^3 + 0x^2 - 5x + 4}\\ \hphantom{x-2 \ )\ } \underline{x^3 - 2x^2}\\ \hphantom{x-2 \ )\ x^3 - {}} 2x^2\\ \hphantom{x-2 \ )\ x^3 - {}} \underline{2x^2 - 4x}\\ \hphantom{x-2 \ )\ x^3 - 2x^2} \ \ {-x}\\ \hphantom{x-2 \ )\ x^3 - 2x^2} \ \ \underline{-x + \,2}\\ \hphantom{x-2 \ )\ x^3 - 2x^2 \, -x} + 2\\ \end{array} }$

Quotient-remainder form:

$x^2+2x-1$, with remainder $2$

Quotient as sum of a polynomial and (proper) rational expression:

$\displaystyle{x^2 + 2x - 1 + \cfrac{2}{x-2}}$

3. $\displaystyle{ \begin{array}[t]{l} \hphantom{x^2 - 3x + 2 \ )\ x^3 + 5x^2 + \ 6}\ x \ + \ 9\\ x^2 - 5x + 7 \ \overline{\smash{\raise{.6ex}{)}}{\vphantom{)}} \ x^3 + 4x^2 - \ \ 6x \ + \ 3}\\ \hphantom{x^2 - 5x + 6 \ )\ } \underline{x^3 - 5x^2 + \ \ 7x}\\ \hphantom{x^2 - 5x + 6 \ )\ x^3 - {}} 9x^2 - 13x\\ \hphantom{x^2 - 5x + 6 \ )\ x^3 - {}} \underline{9x^2 - 45x + 63}\\ \hphantom{x^2 - 5x + 6 \ )\ x^3 - 5x^2 - {}} 32x - 60 \end{array} }$

Quotient-remainder form:

$x+9$, with remainder $32x-60$

Quotient as sum of a polynomial and (proper) rational expression:

$\displaystyle{x+9 + \cfrac{32x-60}{x^2-5x+7}}$

4. $\displaystyle{ \begin{array}[t]{l} \hphantom{3x - 1 \ )\ x^4 + {}}\frac{1}{3}x^3 + \frac{1}{9}x^2 - \frac{17}{27}x - \frac{17}{81}\\ 3x - 1 \ \overline{\smash{\raise{.6ex}{) \ }}{\vphantom{)}} x^4 + \ 0x^3 - 2x^2 + \ \ 0x + \ 1}\\ \hphantom{3x - 1 \ )\ }\underline{x^4 - \frac{1}{3}x^3}\\ \hphantom{3x - 1 \ )\ x^4 - {}} \frac{1}{3}x^3\\ \hphantom{3x - 1 \ )\ x^4 - {}} \underline{\frac{1}{3}x^3 - \frac{1}{9}x^2}\\ \hphantom{3x - 1 \ )\ x^4 - {} \frac{1}{3}x^3} \,{-\frac{17}{9}x^2}\\ \hphantom{3x - 1 \ )\ x^4 - {} \frac{1}{3}x^3} \,\underline{-\frac{17}{9}x^2 + \frac{17}{27}x}\\ \hphantom{3x - 1 \ )\ x^4 - {} \frac{1}{3}x^3 \,-\frac{17}{9}x^2} {-\frac{17}{27}x}\\ \hphantom{3x - 1 \ )\ x^4 - {} \frac{1}{3}x^3 \,-\frac{17}{9}x^2} \underline{-\frac{17}{27}x + \frac{17}{81}}\\ \hphantom{3x - 1 \ )\ x^4 - {} \frac{1}{3}x^3 \,-\frac{17}{9}x^2 -\frac{17}{27}x + }\frac{64}{81}\\ \end{array} }$

Quotient-remainder form:

$\frac{1}{3}x^3 + \frac{1}{9}x^2 - \frac{17}{27}x - \frac{17}{81}$, with remainder $64$

Quotient as sum of a polynomial (with rational coefficients) and (proper) rational expression:

$\frac{1}{3}x^3 + \frac{1}{9}x^2 - \frac{17}{27}x - \frac{17}{81} + \cfrac{64}{3x-1}$

Note, factoring out a $\frac{1}{81}$ will make the answer look a bit "cleaner":

$\cfrac{1}{81} (27x^3 + 9x^2 + 51x - 17) + \cfrac{64}{3x-1}$

2. For each pair of polynomials, express the first as a multiple of the second, plus some remainder as needed.

1. $(x^4 + 5x^3 - 9x + 8); \ (x - 2)$

2. $(x^4 - x^2 + 3x -7); \ (x^2 + x - 3)$

3. $(x^5 - 1); \ (x-1)$

4. $(x^4 - 2x^3 + 4x^2 - 6); \ (x^2 + 2x + 1)$

5. $(x^3+7x^2-5x-8); \ (x-3)$

6. $(x^4-16); \ (x-2)$

1. $(x^3 + 7x^2 + 14x + 19)(x-2) + 46$

2. $(x^2 - x + 3)(x^2 + x - 3) + (-3x+2)$

3. $(x^4 + x^3 + x^2 + x + 1)(x-1)$

4. $(x^2 - 4x + 11)(x^2 + 2x + 1) + (-18x - 17)$

5. $(x^2 + 10x + 25)(x-3) + 67$

6. $(x^3 + 2x^2 + 4x + 8)(x-2)$

3. For each, decide if $(x-c)$ is a factor of the given polynomial without doing any division.

1. $x^3 + 2x^2 + 3x - 6; \ c = 1$

2. $x^4 - x^3 + 2x^2 - 7x - 2; \ c = 2$

3. $x^5 + 1; \ c = -1$

4. $x^3 + 2x + 1; \ c = 1$

1. $(x-1)$ is a factor of $f(x) = x^3 + 2x^2 + 3x - 6$ as $f(1) = 0$

2. $(x-2)$ is a factor of $f(x) = x^4 - x^3 + 2x^2 - 7x - 2$ as $f(2) = 0$

3. $(x+1)$ is a factor of $f(x) = x^5 + 1$ as $f(-1) = 0$

4. $(x-1)$ is not a factor of $f(x) = x^3 + 2x + 1$ as $f(1) \neq 0$

4. Find $(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + x^{n-4}y^3 + x^{n-5}y^4 + x^{n-6}y^5) \div (x^2 + xy + y^2)$

$x^{n-3} + x^{n-6}y^3$

5. Do the division: $\cfrac{a^3 + b^3 + c^3 - 3abc}{a+b+c}$

Given that this division is of polynomials of multiple variables, you may find it challenging to set this up as a typical "long division" problem. Instead, focus on the process for doing division that inspired that "long division" algorithm.