Lesson Notes By Weeks and Term v4 - SHS 1
APPLICATIONS OF ALGEBRA
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APPLICATIONS OF ALGEBRA
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Subject: Additional Mathematics
Class: SHS 1
Term: 1st Term
Week: 18
Grade code: 1.1.2.LI.1
Strand code: 1
Sub-strand code: 2
Content standard code: 1.1.2.CS.1
Indicator code: 1.1.2.LI.1
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATIONS OF ALGEBRA
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In our previous studies, we learned how to combine several simple algebraic fractions into a single, more complex fraction by finding a common denominator. For example, we can easily combine `1/(x-1) + 2/(x+2)` into a single fraction. But what if we are given the complex fraction and need to find the original, simpler fractions that it came from? This process of "un-mixing" a complex fraction is called Partial Fraction Decomposition. It is a powerful tool in algebra that is essential for more advanced topics you will encounter in university, especially in calculus (integration), engineering (circuit analysis), and physics (dynamics).
2.1 What are Rational Functions?
A rational function is a fraction where both the numerator and the denominator are polynomials. It looks like `P(x) / Q(x)`, where `P(x)` and `Q(x)` are polynomials, and `Q(x)` is not zero. Proper Rational Function: The degree (highest power of x) of the numerator `P(x)` is less than the degree of the denominator `Q(x)`. *Example:* `(2x + 3) / (x² - 5x + 6)`. Degree of numerator is 1, degree of denominator is 2. (1 2).
Rule: We can only apply partial fraction decomposition directly to proper rational fractions. If a fraction is improper, we must first perform polynomial long division.
2.2 The Four Cases of Partial Fraction Decomposition