Elena set her pen on the desk. Her palms were damp, but her mind was clear. She had faced the domain restrictions, the partial fraction decomposition, the inverse function trap, the composite’s hidden conditions, and the elegant emptiness of the squared inequality.
She turned the page.
Never. A square of a real number is always ( \geq 0 ). The only time it equals zero is at the roots. So no real ( x ) satisfies ( p(x) < 0 ). core pure -as year 1- unit test 5 algebra and functions
She felt a small smile. But the test wasn't done.
was a curveball—a partial fractions problem disguised as a rational function. Express ( \frac{5x^2 + 4x - 11}{(x-1)(x+2)(x-3)} ) in partial fractions. Her pen flew. She set up the identity: ( 5x^2 + 4x - 11 \equiv A(x+2)(x-3) + B(x-1)(x-3) + C(x-1)(x+2) ). She chose the cover-up rule for speed: ( x=1 ) gave ( A = 1 ). ( x=-2 ) gave ( B = -1 ). ( x=3 ) gave ( C = 5 ). Elena set her pen on the desk
Roots: ( x = 2 ) and ( x = -2 ), both repeated (multiplicity 2). The inequality ( p(x) < 0 ) asked: when is a square less than zero?
Unit Test 5 wasn't just about algebra. It was about precision. About checking every assumption. About remembering that a square can never be negative. She turned the page
The answer formed: ( \frac{1}{x-1} - \frac{1}{x+2} + \frac{5}{x-3} ). Clean. Elegant.