Lesson goal: Partial Fractions

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Partial fraction decomposition is a technique used in algebra and calculus to break down a rational expression into simpler fractions: $$\frac{P(x)}{Q(x)} = \frac{A}{x - r_1} + \frac{B}{x - r_2}$$ In the 1966 MAA High School Mathematics Contest (Problem #7), students were given the identity: $$\frac{35x - 29}{x^2 - 3x + 2} = \frac{N_1}{x - 1} + \frac{N_2}{x - 2}$$ and asked to find the product $N_1 \cdot N_2$. First, we factor the quadratic denominator $x^2 - 3x + 2 = (x - 1)(x - 2)$ using CodeByMath's symbolic factor function. Multiplying both sides of the identity by $(x - 1)(x - 2)$ clears the denominators: $$35x - 29 = N_1(x - 2) + N_2(x - 1)$$ Because this equation holds for all values of $x$, we can substitute the roots to solve for $N_1$ and $N_2$ directly:
  • Setting $x = 1$: $35(1) - 29 = N_1(1 - 2) \implies 6 = -N_1 \implies N_1 = -6$
  • Setting $x = 2$: $35(2) - 29 = N_2(2 - 1) \implies 41 = N_2 \implies N_2 = 41$
In this lesson, we will use symbolic algebra and computer calculation to solve and verify partial fractions!
denom = factor("x^2 - 3*x + 2")
check = expand("-6*(x - 2) + 41*(x - 1)")
product = N1 * N2
Move the mouse over a dotted box for more information.

Notice how evaluating the equation at $x = 1$ makes the $N_2(x - 1)$ term vanish completely, while setting $x = 2$ eliminates the $N_1(x - 2)$ term! This is often called the cover-up method.

Now you try. Set N1 = -6, N2 = 41, and product = N1 * N2. Run the code to verify the partial fraction values and find the numerical product!

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