WHAT IS THE COEFFICIENT OF x3 IN THE EXPANSION OF (2 − x/2)6?
The coefficient is −20. The powers of 2 and of ½ cancel exactly, so the whole term collapses to −C(6,3), and the minus sign survives because the x term is being raised to an odd power.
What is the coefficient of x3 in the binomial expansion of (2 − x⁄2)6?
Write the single term you want rather than the whole expansion. You need one binomial coefficient, one power of 2 and one power of one half, and two of those three cancel.
- the x3 term is C(6,3) × 23 × (−x⁄2)3
- C(6,3) = 20
- 23 × (−½)3 = 8 × (−⅛) = −1
- coefficient = 20 × (−1) = −20
+20 is what you get if you drop the sign The second term of the bracket is negative and it is being cubed, so the term stays negative. An even power would have cleared it; an odd one does not.
−160 is what you get if you expand (2 − x)⁶ by mistake and never halve the x. That gives 20 × 8 × (−1) = −160. Whenever a bracket carries a fraction, the fraction is there precisely because the question wants to see whether you raise it to the power as well.
Look at 2 and ½ before you multiply anything. They are reciprocals, so 2³ × (½)³ = 1 for any power you are asked about, and the coefficient of every term in this expansion is just ±C(6, r). That turns the question into "what is C(6,3)?" and you are done.
THE GENERAL RULE
A binomial question that asks for one coefficient is testing whether you can isolate a single term instead of expanding seven of them. Write the general term first, then look at the numbers in it for anything that cancels — reciprocals, powers of the same base, factors that divide out. In a module that gives you roughly 89 seconds a question, spotting that the arithmetic collapses is worth more than being quick at arithmetic.
Updated · written to the published ESAT content specification