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: 17
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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This lesson introduces students to rational functions, which are essentially fractions where the numerator and denominator are polynomials. We will first learn how to add, subtract, multiply, and divide these functions, much like we do with ordinary fractions. Then, we will learn a powerful technique called "partial fraction decomposition," which is like breaking down a complex fraction into simpler, more manageable parts. This skill is not just an algebraic exercise; it is a fundamental tool used in advanced mathematics (like calculus), engineering, and economics to solve complex problems.
This section is divided into two parts. Part A covers the basic operations, and Part B covers partial fraction decomposition. Part A: Basic Operations on Rational Functions What is a Rational Function? A rational function is a function that can be written as the ratio of two polynomials, `P(x)` and `Q(x)`. `f(x) = P(x) / Q(x)` ...where `Q(x)` is not the zero polynomial. Think of it like a numerical fraction (e.g., 3/4), but with variables. Example: `f(x) = (x + 2) / (x² - 9)` is a rational function. The most important rule is that the denominator cannot be zero. The values of `x` that make the denominator zero are excluded from the domain of the function. For `f(x)` above, `x² - 9 = 0` when `x = 3` or `x = -3`. So, `x` cannot be 3 or -3. Simplifying Rational Functions The key to simplifying is factorisation. We factorise both the numerator and the denominator and then cancel out any common factors.