KCSE 2025 Maths P1 Q7 — Simplify an Algebraic Fraction

KCSE 2025 Form 4 Algebra

Published

The Question

“Simplify the algebraic fraction with numerator x squared minus 4y squared and denominator x squared plus 4xy plus 4y squared.”

1

Plan the approach

To simplify any algebraic fraction, factorise the numerator and denominator as fully as possible, then cancel any factor that appears in both. Do not try to cancel individual terms; you can only cancel whole factors.

x2−4y2x2+4xy+4y2\frac{x^{2} - 4y^{2}}{x^{2} + 4xy + 4y^{2}}
2

Factorise the numerator

The top is a difference of two squares: x squared is a square and 4y squared is the square of 2y. A difference of two squares a squared minus b squared factorises into two brackets, one with a minus and one with a plus. Here a is x and b is 2y.

x2−4y2=x2−(2y)2x^{2} - 4y^{2} = x^{2} - (2y)^{2}
x2−(2y)2=(x−2y)(x+2y)x^{2} - (2y)^{2} = (x - 2y)(x + 2y)
3

Factorise the denominator

The bottom is a perfect square trinomial. Expanding x plus 2y all squared gives x squared, plus twice the product of x and 2y which is 4xy, plus 2y all squared which is 4y squared. This matches the denominator exactly, so it factorises to a single squared bracket.

(x+2y)2=x2+4xy+4y2(x + 2y)^{2} = x^{2} + 4xy + 4y^{2}
x2+4xy+4y2=(x+2y)(x+2y)x^{2} + 4xy + 4y^{2} = (x + 2y)(x + 2y)
4

Cancel the common factor

Write the factorised form of the whole fraction. The factor x plus 2y appears once on the top and twice on the bottom, so cancel one copy from each. What remains is the simplified fraction.

(x−2y)(x+2y)(x+2y)(x+2y)=x−2yx+2y\frac{(x - 2y)(x + 2y)}{(x + 2y)(x + 2y)} = \frac{x - 2y}{x + 2y}

Final Result

The fraction simplifies to (x - 2y) divided by (x + 2y).

Why this method works

The method works because a fraction only shrinks to lowest terms when identical whole factors sit in both the numerator and denominator, since any non-zero quantity divided by itself equals one. Factorising exposes those hidden common factors: the difference of two squares reveals the bracket (x + 2y), and the same bracket is buried inside the perfect square on the bottom. Once both are written as products, cancelling that shared bracket is just dividing top and bottom by the same amount, which never changes the value of the fraction.

✓

Substitute x = 3 and y = 1: original gives (9 - 4) / (9 + 12 + 4) = 5/25 = 0.2, and the answer gives (3 - 2)/(3 + 2) = 1/5 = 0.2, so they agree.