Simplify the rational expression x2 − 7x + 125x2 + 5x − 100, assuming that 5x2 + 5x − 100 ≠ 0.
Step-by-step solution
To find: The simplest form
Idea: Factorise the top and the bottom completely, then cancel any factor that appears in both. Cancelling is allowed only because that factor is not zero.
- Numerator: x2 − 7x + 12. Need a + b = −7, ab = 12 → −3 and −4. So x2 − 7x + 12 = (x − 3)(x − 4).1 mark
- Denominator: all terms are multiples of 5: 5x2 + 5x − 100 = 5(x2 + x − 20). Need a + b = 1, ab = −20 → 5 and −4. So = 5(x − 4)(x + 5).1 mark
- x2 − 7x + 125x2 + 5x − 100 = (x − 4)(x − 3)5(x − 4)(x + 5)½ mark
- Since the denominator is not 0, x − 4 ≠ 0, so we may cancel it: = x − 35(x + 5).½ mark
Check: x = 1: original = (1 − 7 + 12)/(5 + 5 − 100) = 6/(−90) = −115; simplified = (1 − 3)/(5 × 6) = −2/30 = −115 ✓.
Answer to write in the exam
x2 − 7x + 12 = (x − 3)(x − 4)
5x2 + 5x − 100 = 5(x2 + x − 20) = 5(x − 4)(x + 5)
x2 − 7x + 125x2 + 5x − 100 = (x − 3)(x − 4)5(x − 4)(x + 5)
∴ x2 − 7x + 125x2 + 5x − 100 = x − 35(x + 5)
Common mistakes that cost marks
- Cancelling terms instead of factors, e.g. crossing out x2 from top and bottom. Only whole factors that multiply can be cancelled.
- Forgetting the 5 that was taken out of the denominator.
- Getting the signs in 5 and −4 the wrong way round: (x + 4)(x − 5) gives −x, not +x.
How this can come in the exam
x2 − 9x2 + 5x + 6 (denominator ≠ 0) simplifies to
- x − 3x + 2
- x + 3x + 2
- x − 3x + 3
- −95x + 6
Show answer
(A) x − 3x + 2
(x − 3)(x + 3) / ((x + 2)(x + 3)) = (x − 3)/(x + 2).
Simplify 2x2 − 8x2 + x − 6, assuming the denominator is not zero.
Show answer
Top: 2(x2 − 4) = 2(x − 2)(x + 2). Bottom: (x + 3)(x − 2) (1 mark). Cancel (x − 2): 2(x + 2)x + 3 (1 mark).Try one yourself
Simplify x2 + 3x − 103x2 − 12, assuming the denominator is not zero.
Show answer
(x + 5)(x − 2) / (3(x − 2)(x + 2)) = x + 53(x + 2).
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