Algebra

Polynomial long division

\(\displaystyle \frac{f(x)}{g(x)}\)

1. search biggest exponents in dividend and divisor, e.g. \(ax^n\) and \(bx^m\)

2. if \(n \geq m\):

   • add \(\displaystyle \frac{a}{b}x^{n-m}\) to the result

   else:

   • you can’t divide any further
   • add \(\displaystyle \frac{rest}{g(x)}\) to the result
   • finished

3. subtract \(g(x) \cdot \displaystyle \frac{a}{b}x^{n-m}\) from \(f(x)\)

4. repeat with result of 3.

Examples

[1]

[2]

System of linear equations [3]

Solve with TI-84 Plus [4]

1. 2nd, MATRIX
2. navigate to EDIT with left/right arrows
3. choose the matrix to edit with the up/down arrows + ENTER or a number
4. set dimensions to \(X\times X+1\) when you have \(X\) variables
5. type in the coefficents of your equations in the last column write the RHS of your equations
6. 2nd, QUIT
7. 2nd, MATRIX
8. navigate to MATH with left/right arrows
9. choose rref( with ALPHA, B or the up/down arrows + ENTER
10. 2nd, MATRIX
11. choose the matrix you edited with the up/down arrows + ENTER or a number
12. ENTER

Solve by hand

1. take any equation of your system
2. isolate any of the variables in the equation
3. substitute it into the other equations
4. repeat until any variable’s value is found
5. substitute back until you have the values of all variables

Examples

\[\begin{split}\begin{alignedat}{7} 3x &\ & + &\ & 2y &\ & - &\ & z &\ & = &\ & 1 \\ 2x &\ & - &\ & 2y &\ & + &\ & 4z &\ & = &\ & -2 \\ -x &\ & + &\ & {\tfrac {1}{2}}y &\ & - &\ & z &\ & = &\ & 0 \\ \end{alignedat}\end{split}\]

Solution: \(x = 1, y = -2, z = -2\)

[1]	https://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Polynomdivision_1.svg/2880px-Polynomdivision_1.svg.png

[2]	https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Polynomdivision_3.svg/2880px-Polynomdivision_3.svg.png

[3]	https://en.wikipedia.org/wiki/System_of_linear_equations

[4]	https://education.ti.com/en/products/calculators/graphing-calculators/ti-84-plus