Adding/Subtracting Fractions

Are you missing any steps?

$\frac{1}{2} + \frac{3}{5}$

$=\frac{1}{2}\cdot \frac{5}{5}+\frac{3}{5}\cdot \frac{2}{2}$
$=\frac{5}{10}+\frac{6}{10}$
$=\frac{5+6}{10}$
$=\frac{11}{10}$

Adding/Subtracting Fractions

We can say that
$\frac{1}{2} + \frac{3}{5}=\frac{11}{10}$
and
$\frac{11}{10}=\frac{1}{2} + \frac{3}{5}$.

Adding/Subtracting Fractions

Subtract $\frac{1}{x+2}-\frac{3}{x-5}$.

Write out all the steps in detail. Don't assume anything!

Adding/Subtracting Fractions

Are you missing any steps?

$\frac{1}{x+2}-\frac{3}{x-5}$

$=\frac{1}{x+2}\cdot \frac{x-5}{x-5}-\frac{3}{x-5}\cdot \frac{x+2}{x+2}$

$=\frac{1(x-5)}{(x+2)(x-5)}-\frac{3(x+2)}{(x-5)(x+2)}$
$=\frac{1(x-5)-3(x+2)}{(x+2)(x-5)}$
$=\frac{x-5-3x-6}{(x+2)(x-5)}$

$=\frac{-2x-11}{(x+2)(x-5)}$

Adding/Subtracting Fractions

We can say that
$\frac{1}{x+2}-\frac{3}{x-5}=\frac{-2x-11}{(x+2)(x-5)}$
and

$\frac{-2x-11}{(x+2)(x-5)}=\frac{1}{x+2}-\frac{3}{x-5}$.

Adding/Subtracting Fractions

What if you were given 

$\frac{-2x-11}{(x+2)(x-5)}$ and need to find that it came from $\frac{1}{x+2}-\frac{3}{x-5}$?

What would $\frac{x+1}{(x-10)(x+4)}$ split into?

Integrating Fractions

If we can find a way to 'decompose' a complex fraction into its component 'partial fractions,' we can integrate those smaller fractions much more quickly.

Classifying Problems

The denominator in $\frac{3 x+11}{(x-3)(x+2)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{2+x^4}{x(x^2+9)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{2+x^4}{x(x^2+9)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{3 x+11}{(x+2)(x+2)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{3 x+11}{(x+2)(x+2)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{3 x+11}{x^2+6x+9}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{3 x+11}{x^2+6x+9}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{4 x-11}{x^3-9 x^2}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{4 x-11}{x^3-9 x^2}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

The denominator in $\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}$ has:

A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors

Classifying Problems

$\frac{3 x+11}{(x-3)(x+2)}$ would decompose to

A. $\frac{A}{x-3}+\frac{B}{x+2}$

B. $\frac{A}{x-3}+\frac{Bx}{x+2}$
C. $\frac{A}{x-3}+\frac{Bx+C}{x+2}$
D. $\frac{Ax+B}{x-3}+\frac{Cx+D}{x+2}$

Classifying Problems

$\frac{2+x^4}{x(x^2+9)}$ would decompose to

A. $\frac{A}{x}+\frac{B}{x^2+9}$
B. $\frac{Ax+B}{x}+\frac{C}{x^2+9}$
C. $\frac{A}{x}+\frac{Bx+C}{x^2+9}$
D. $\frac{Ax+B}{x}+\frac{Cx+D}{x^2+9}$

Classifying Problems

$\frac{2+x^4}{x(x^2+9)}$ would decompose to

A. $\frac{A}{x}+\frac{B}{x^2+9}$
B. $\frac{Ax+B}{x}+\frac{C}{x^2+9}$
C. $\frac{A}{x}+\frac{Bx+C}{x^2+9}$
D. $\frac{Ax+B}{x}+\frac{Cx+D}{x^2+9}$

Classifying Problems

$\frac{3 x+11}{(x+2)(x+2)}$ would decompose to

A. $\frac{A}{x+2}+\frac{B}{x+2}$

B. $\frac{A}{x+2}+\frac{B}{(x+2)^2}$
C. $\frac{A}{x+2}+\frac{Bx+C}{x+2}$
D. $\frac{Ax+B}{x+2}+\frac{Cx+D}{(x+2)^2}$

Classifying Problems

$\frac{3 x+11}{(x+2)(x+2)}$ would decompose to

A. $\frac{A}{x+2}+\frac{B}{x+2}$

B. $\frac{A}{x+2}+\frac{B}{(x+2)^2}$
C. $\frac{A}{x+2}+\frac{Bx+C}{x+2}$
D. $\frac{Ax+B}{x+2}+\frac{Cx+D}{(x+2)^2}$

Classifying Problems

$\frac{3 x+11}{x^2+6x+9}$ would decompose to

A. $\frac{A}{x+3}+\frac{B}{x+3}$

B. $\frac{A}{x+3}+\frac{B}{(x+3)^2}$
C. $\frac{A}{x+3}+\frac{Bx+C}{x+3}$
D. $\frac{Ax+B}{x+3}+\frac{Cx+D}{(x+3)^2}$

$\frac{3 x+11}{x^2+6x+9}$ would decompose to

A. $\frac{A}{x+3}+\frac{B}{x+3}$

B. $\frac{A}{x+3}+\frac{B}{(x+3)^2}$
C. $\frac{A}{x+3}+\frac{Bx+C}{x+3}$
D. $\frac{Ax+B}{x+3}+\frac{Cx+D}{(x+3)^2}$

Classifying Problems

Classifying Problems

$\frac{4 x-11}{x^3-9 x^2}$ would decompose to

A. $\frac{A}{x^2}+\frac{B}{x-9}$

B. $\frac{Ax+B}{x^2}+\frac{C}{x-9}$
C. $\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-9}$
D. $\frac{A}{x}+\frac{Bx+C}{x^2}+\frac{D}{x-9}$

Classifying Problems

$\frac{4 x-11}{x^3-9 x^2}$ would decompose to

A. $\frac{A}{x^2}+\frac{B}{x-9}$

B. $\frac{Ax+B}{x^2}+\frac{C}{x-9}$
C. $\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-9}$
D. $\frac{A}{x}+\frac{Bx+C}{x^2}+\frac{D}{x-9}$

Classifying Problems

$\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}$ would decompose to

A. $\frac{A}{z-6}+\frac{B}{z^2+4}$

B. $\frac{A}{z-6}+\frac{Bx+C}{z^2+4}$
C. $\frac{Ax+B}{z-6}+\frac{Cx+D}{z+2}+\frac{Ex+F}{z-2}$
D. $\frac{A}{z-6}+\frac{B}{z+2}+\frac{C}{z-2}$

$\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}$ would decompose to

A. $\frac{A}{z-6}+\frac{B}{z^2+4}$

B. $\frac{A}{z-6}+\frac{Bx+C}{z^2+4}$
C. $\frac{Ax+B}{z-6}+\frac{Cx+D}{z+2}+\frac{Ex+F}{z-2}$
D. $\frac{A}{z-6}+\frac{B}{z+2}+\frac{C}{z-2}$

Classifying Problems

Classifying Problems

$\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}$ would decompose to

A. $\frac{A}{x-1}+\frac{B}{x^2+4}$

B. $\frac{A}{x-1}+\frac{Bx+C}{(x^2+4)^2}$
C. $\frac{A}{x-1}+\frac{Bx+C}{x^2+4}+\frac{Ex+F}{(x^2+4)^2}$
D. $\frac{A}{x-1}+\frac{B}{x^2+4}+\frac{C}{(x^2+4)^2}$

Classifying Problems

$\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}$ would decompose to

A. $\frac{A}{x-1}+\frac{B}{x^2+4}$

B. $\frac{A}{x-1}+\frac{Bx+C}{(x^2+4)^2}$
C. $\frac{A}{x-1}+\frac{Bx+C}{x^2+4}+\frac{Ex+F}{(x^2+4)^2}$
D. $\frac{A}{x-1}+\frac{B}{x^2+4}+\frac{C}{(x^2+4)^2}$