SSC CGL 2024 Tier-I · 9 September 2024, Shift 1 · Quantitative Aptitude · Question 21 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 6 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

Simplify [(x³ + 15x² + 75x + 125) ⁄ (x² − 25)] (x − 5).

Answer

Why

Correct — C. The stem and all four options are images. The stem reads: Simplify (x³ + 15x² + 75x + 125)/(x² − 25) × (x − 5). Option (c) shows x² + 10x + 25.

Factorise both parts.
x³ + 15x² + 75x + 125 = (x + 5)³, matching a³ + 3a²b + 3ab² + b³ with a = x and b = 5
x² − 25 = (x + 5)(x − 5)

So the expression is (x + 5)³ ÷ [(x + 5)(x − 5)] × (x − 5).

The (x − 5) cancels against the multiplier and one (x + 5) cancels below, leaving (x + 5)² = x² + 10x + 25 → option (c).

Check at x = 1: 216/(−24) × (−4) = 36, and 1 + 10 + 25 = 36.

Why the others are wrong

(a)
The middle coefficient is wrong: (x + 5)² = x² + 10x + 25, not x² + 5x + 25. At x = 1 option (a) is 31 while the expression is 36.
(b)
This is the denominator copied down, x² − 25, not a simplified result. At x = 1 it is −24 against the expression's 36.
(d)
Squaring a binomial keeps the cross term 2·x·5, so (x + 5)² is not x² + 25. At x = 1 option (d) is 26, not 36.

Concept

Two standard identities do all the work here.

The cubic is a perfect cube: a³ + 3a²b + 3ab² + b³ = (a + b)³, and with a = x, b = 5 the coefficients 1, 15, 75, 125 fall straight out.

The quadratic below is a difference of squares: x² − 25 = (x + 5)(x − 5).

Once both are factored the cancellation is mechanical and nothing has to be expanded or divided out long-hand.

Strictly the expression is undefined at x = 5 and x = −5, where the denominator is zero. SSC ignores that and asks only for the simplified form, which is standard for this question type.

Key facts

(a + b)³ = a³ + 3a²b + 3ab² + b³
With a = x and b = 5 this gives x³ + 15x² + 75x + 125 = (x + 5)³
a² − b² = (a + b)(a − b), so x² − 25 = (x + 5)(x − 5)
(x + 5)² expands to x² + 10x + 25

Study next

Recognising (a + b)³ and (a − b)³ from their coefficient pattern 1, 3, 3, 1Factorising by difference of squares and by sum or difference of cubesSubstituting a convenient value of x to test a simplified algebraic answer

Common traps

Reading x³ + 125 out of the cubic and applying the sum-of-cubes factorisation instead
Dropping the cross term and writing (x + 5)² as x² + 25
Cancelling (x − 5) against the wrong bracket and leaving the denominator behind

A simplify-this-rational-expression item also appears at 09 Sep 2024, 09:00, Quant Q.15, which asks for (x² − 9)/(x + 3).

Both are decided by spotting a standard identity, never by long division. Checking your answer at x = 1 takes about five seconds and catches every wrong option here.

Related PYQs

No directly related past PYQ was found.