Partial Fractions

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.4 – Integration of Rational Functions by Partial Fractions

Prerequisite Knowledge

This section contains the specific content from previous courses that you may need for CPT_06_Partial_Fractions.

Algebra

Factoring Expressions

• Difference of Two Squares ${\displaystyle (a^2{x}^2-b^2)=(ax+b)(ax-b)}$$\displaystyle \left(a^2x^2-b^2\right)=\left(ax+b\right)\left(ax-b\right)$
• Sum of Two Squares [NOT factorable using Real numbers, $\mathbb{R}$$\mathbb{R}$
• Difference of Two Cubes ${\displaystyle (a^3{x}^3-b^3)=(ax-b)(a^2{x}^2+abx+b^2)}$$\displaystyle \left(a^3x^3-b^3\right)=\left(ax-b\right)\left(a^2x^2+abx+b^2\right)$
• Sum of Two Cubes ${\displaystyle (a^3{x}^3+b^3)=(ax+b)(a^2{x}^2-abx+b^2)}$$\displaystyle \left(a^3x^3+b^3\right)=\left(ax+b\right)\left(a^2x^2-abx+b^2\right)$
• Factor Using Distributive Property ${\displaystyle x^2-3x=x(x-3)}$$\displaystyle x^2-3x=x(x-3)$
• Factor Quadratic Expressions ${\displaystyle a{x}^2+bx+c=(x+s)(x+t)}$$\displaystyle ax^2 + bx + c = (x + s)(x + t)$, where $(s)(t)=c$$(s)(t) = c$ and $s+t=b$$s + t = b$
• Factor Quadratic-like Expressions ${\displaystyle a{x}^{2n}+b{x}^n+c}$$\displaystyle ax^{2n} + bx^n + c$. Let $u=x^n$$u=x^n$. Then the expression becomes ${\displaystyle a{u}^2+bu+c=(u+s)(u+t)}$$\displaystyle au^{2} + bu + c = (u + s)(u + t)$ where $(s)(t)=c$$(s)(t) = c$ and $s+t=b$$s + t = b$. Finally, use the let statement to convert from $u$$u$ to $x$$x$.
• Factor by Grouping ${\displaystyle a{x}^3+b{x}^2+cx+d}$$\displaystyle ax^3 + bx^2 + cx + d$, where $c=sa$$c = sa$ and $d=sb$$d = sb$ for some $s$$s$. Then $(a{x}^3+b{x}^2)+(cx+d)=x^2(ax+b)+s(ax+b)=(ax+b)(x^2+s)$$\left(ax^3 + bx^2\right) + (cx + d) = x^2 (ax + b) + s (ax + b) = (ax + b)\left(x^2 + s\right)$

Use Technology to Factor Expressions

NOTE: On Assessments, you may use of technology to factor expressions without showing any work.

Mathematica

1
(* Factor 8x^3-1 *)
2
Factor[8x^3-1]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. Factor[ ] has one arguments:

   1. The expression to be factored.

3. For help on using the Factor[ ] function:

   1. In Mathematica, execute the code: $\text{?Factor}$$\text{?Factor}$
   2. Click on $\vee$$\or$ near the bottom-left of output
   3. Click on local
   4. Read how to use the Factor[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.
2. Python requires $\ast$$*$ for explicit multiplication.
3. Python requires $\ast \ast$$**$ for exponentiation, e.g., $x^3$$x^3$ is $\text{x**3}$$\text{x**3}$.
4. Remember to use parens $($$\left(\right.$ and $)$$\left.\right)$​ to group factors consisting of multiple terms.
5. You can use pprint(factor(8x*3-1)) for pretty print.
6. To execute code, press the “View the result” button:

Rational Exponents

${\displaystyle \sqrt[b]{x^a}=x^{\frac{a}{b}}}$$\displaystyle\sqrt[b]{x^a}=x^{\frac{a}{b}}$

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Created: Monday, 06 September 2021 - 12:36 (EDT)