Adding/Subtracting Fractions

Add \(\frac{1}{2}+\frac{3}{5}\).

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

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

What is \(\int \frac{1}{x+2}dx\)?

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 \(\frac{3 x+11}{(x-3)(x+2)}\) 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

\(\frac{3 x+11}{(x-3)(x+2)}\) 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. \(\frac{A}{x-1}+\frac{B}{x^2+4}+\frac{C}{(x^2+4)^2}\)