Find the coefficient of the given term in the binomial expansion. x^10 ​term, (x-5)^10

Question

Find the coefficient of the given term in the binomial expansion.

x^10 ​term, (x-5)^10

Answer 1

In order to find the coefficient of just one term in a binomial expansion, we can use the general term from the binomial theorem instead of multiplying the whole thing out ten times. The general term for (a + b) raised to the n looks like this:

C(n, k) a^n-k b^k

where k counts from 0 up to n, and C(n, k) is the binomial coefficient that you can get off Pascal's triangle or with the nCr button on your calculator.

For your problem, (x - 5)¹⁰, we have n = 10, a = x, and b = -5. Make sure to keep that minus sign attached to the 5 instead of treating it as subtraction you will handle later, because that is where most of the mistakes on these problems come from. Putting those in, our general term becomes:

C(10, k) x^10-k (-5)^k

Remember, our purpose is to find the x¹⁰ term, so we need the exponent on x to come out to 10. The exponent on x is 10 - k, so we can set up a small equation and solve for k:

10 - k = 10, so k = 0

That tells us the x¹⁰ term is the very first term of the expansion. Now we substitute k = 0 back into the general term:

C(10, 0) x¹⁰ (-5)⁰ = (1)(x¹⁰)(1) = x¹⁰

If you notice, two things both turned into 1 there. C(10, 0) is 1 because there is only one way to choose nothing out of ten things, and (-5)⁰ is 1 because any number other than zero raised to the zero power is 1. So the 5 never even gets into this term, which is why the answer looks surprisingly plain.

That also makes sense if you picture actually multiplying (x - 5) by itself ten times. To end up with x¹⁰ you would have to pick the x out of every single one of the ten factors, and there is exactly one way to do that, so nothing gets multiplied onto it. The -5 only starts showing up in the terms where you picked it from at least one factor, and all of those have a lower power of x. For example the next term, where k = 1, is C(10, 1) x⁹ (-5)¹ = (10)(x⁹)(-5) = -50x⁹, so there you can see the 5 and the minus sign finally appear.

One last thing to keep in mind is that they asked for the coefficient, which is just the number sitting in front of the power of x. So the answer is the number 1, not x¹⁰. Also, since the leading term never picks up the constant, the coefficient of the highest power in any expanded (x + c)ⁿ is always going to be 1, which is a handy thing to recognize so you do not have to work it out next time.

So, our final answer is 1.

Answer 2

Coefficient of term with x¹⁰ in this expansion C₁₀⁰ = 1