Try to simplify the following rational expression:
36s2 − 12st + t2t2 + 2ts − 48s2 = (6s − t)2(___ + ___)(___ + ___) .
(Hint: Factor t2 + 2ts − 48s2 and simplify the rational expressions assuming that t2 + 2ts − 48s2 ≠ 0).
Step-by-step solution
To find: The factors of the denominator, and the simplest form
Idea: The numerator is a perfect square: 36s2 − 12st + t2 = (6s)2 − 2(6s)(t) + t2 = (6s − t)2. Factor the denominator as (t + a)(t + b) with a + b = 2s and ab = −48s2, then look for a common factor. Note that (6s − t) and (t − 6s) are negatives of each other, but their squares are equal.
- Numerator: 36s2 − 12st + t2 = (6s − t)2 (identity (a − b)2 with a = 6s, b = t).½ mark
- Denominator: t2 + 2st − 48s2. Need two terms with sum 2s and product −48s2: 8s and −6s (8 − 6 = 2, 8 × (−6) = −48).½ mark
- So t2 + 2ts − 48s2 = (t + 8s)(t + (−6s)) = (t + 8s)(t − 6s). These fill the blanks.½ mark
- (6s − t)2 = (t − 6s)2 = (t − 6s)(t − 6s), because squaring removes the sign.½ mark
- (t − 6s)(t − 6s)(t + 8s)(t − 6s). The denominator is not 0, so t − 6s ≠ 0 and we can cancel it.½ mark
- = t − 6st + 8s½ mark
Check: s = 1, t = 2: original = (36 − 24 + 4)/(4 + 4 − 48) = 16/(−40) = −25; simplified = (2 − 6)/(2 + 8) = −4/10 = −25 ✓.
Answer to write in the exam
36s2 − 12st + t2 = (6s − t)2 = (t − 6s)2 [a2 − 2ab + b2 = (a − b)2]
t2 + 2ts − 48s2 = t2 + 8ts − 6ts − 48s2 = (t + 8s)(t − 6s)
36s2 − 12st + t2t2 + 2ts − 48s2 = (t − 6s)2(t + 8s)(t − 6s)
∴ 36s2 − 12st + t2t2 + 2ts − 48s2 = t − 6st + 8s
Common mistakes that cost marks
- Cancelling (6s − t) with (t − 6s) as if they were equal. They are negatives of each other; work with (t − 6s)2 instead.
- Factoring the denominator as (t − 8s)(t + 6s). That gives −2ts in the middle, not +2ts.
- Cancelling the squared terms 36s2 and 48s2 directly. Only common factors can be cancelled.
How this can come in the exam
t2 + 2ts − 48s2 factorises as
- (t − 8s)(t + 6s)
- (t + 8s)(t − 6s)
- (t + 12s)(t − 4s)
- (t + 16s)(t − 3s)
Show answer
(B) (t + 8s)(t − 6s)
8s + (−6s) = 2s and 8s × (−6s) = −48s2.
Simplify a2 − 10ab + 25b2a2 − 2ab − 15b2, assuming the denominator is not zero.
Show answer
Top = (a − 5b)2; bottom = (a − 5b)(a + 3b) (1 mark). Cancel (a − 5b): a − 5ba + 3b (1 mark).Try one yourself
Simplify 4x2 + 4xy + y24x2 − y2, assuming the denominator is not zero.
Show answer
(2x + y)2 / ((2x + y)(2x − y)) = 2x + y2x − y.
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