Gerard M. answered • 02/20/22
Enthusiastic Math Tutor With 4 Years Experience
Binomials raised to higher and higher powers have a complex, yet orderly expansion. You may know this already, but the coefficients of each term in a binomial expansion are determined by the "choose" function. Here's an example:
(a+b)4 = (4C0)a4 + (4C1)a3b1 + (4C2)a2b2 + (4C3)a1b3 + (4C4)b4 = 1a4 + 4a3b1 + 6a2b2 + 4a1b3 + 1b4
Now, you're given the 5th term to some expansion n of (ax + by)n, which is 560x3y4. We can tell that n for this expansion is 7, because there's a count of 3 x's and 4 y's (notice in the above expansion, the degree of all terms are the same, 4). Given this, we can figure out what the 5th coefficient of (x+y)7 is by using the "choose" function, 7C4 = 35. (Edit: We use 4 instead of 5 in the choose function because, as you can see in the expansion above, the first term starts from 4C0, so it's like it's one behind. You may even count that the fifth term in that expansion has coefficient 4C4.)
However, this (35) is the 5th coefficient for (x+y)7. What about (ax + by)7? Well if you expand the first couple of degrees, you'll notice that the number of a's in each term is equal to the number of x's, and likewise for y. Here's a demonstration:
(ax + by)2 = a2x2 + 2axby + b2y2
(ax + by)3 = a3x3 + 3a2x2by + 3axb2y2 + b3y3
(ax + by)7 = a7x7 +⋅ ⋅ ⋅ + (7C3)a4x4b3y3 + (7C4)a3x3b4y4 + ⋅ ⋅ ⋅ + b7y7
^ Recall that this is equal to 35
What this means is that the 5th term of our 7 degree binomial expansion is equivalent to 35a3b4x3y4, which, as given by the problem, is also equivalent to 560x3y4. Since they're equivalent, we can set them equal and try to solve for a and b:
We can't really solve this last part analytically, but knowing that a and b are integers, we should be able to guess our way to the answer. First of all, a has to be positive, since it has an odd power and b is an even power. Through some observation, you should be able to find that a has to equal 1, leaving b to be the 4th root of 16, which is ±2. Yes, there are multiple solutions, don't forget the negative root!
In summary: a = 1, b = ±2. But just for fun, consider the fact that a variable raised to some power has that many solutions to the equation. We only found two though, where are the missing two? They're imaginary, of course! Don't leave too quickly though, recall that i = √(-1), and what happens when i is raised to the 4th power? i4 = i2 * i2 = √(-1)2 * √(-1)2 = -1 * -1 = 1. So if we let b = ±2i, b4 = (±2)4i4 = 16 * 1 = 16, which means ±2i are the last two solutions to our equation!
Hope you find this last fact interesting, and if you need any further clarification, feel free to leave a comment!
Gerard M.
Let's take it piece-by-piece. The 35 comes from the fact that the 5th term of a 7 degree expansion can be calculated using the choose function, nCp, specifically (7)C(4), which is 35 (that part is explained in paragraph 2, study the (a+b)^4 expansion and look for patterns to help understand). We know the term contains (x^3)(y^4) partially because its given to us, but also because that's the pattern in binomial expansion. The x power starts from the degree 7, and goes down to 6, 5, 4, 3 by the 5th term, whereas the y term increases from 0 to 1, 2, 3, 4. The last bit, (a^3)(b^4), is a bit tricky, but its basically the same as the xy bit, because remember, the original binomial is (ax + by). If you multiply it out, you'll find that whenever x multiplies x, a multiplies a, and whenever y multiplies y, b multiplies b. The effect is that all a's have the same power as x, and all b's have the same power as y. I hope that makes sense, it's difficult to elaborate as a comment. I'll also try making some minor adjustments to my answer to make things clear :)02/20/22
Ariel G.
Hi Gerard, thank you so much for your detailed explanation, I never see any tutors put in this much dedication into their responses. Now I fully understood why we need to divide, but I was a little confused on the expansion part. I am not sure why the 5th term of the 7 degree binomial expansion is equivalent to 35a^3b^4x^3y^4?02/20/22