For example, i'm not suppose to add 4x^3+6x^2. Why can't it be 8x^5? I thought if you are add like terms (x), then you can just add the exponents. Can someone explain this to me?
For example, i'm not suppose to add 4x^3+6x^2. Why can't it be 8x^5? I thought if you are add like terms (x), then you can just add the exponents. Can someone explain this to me?
You can see that 4x^3 + 6x^3 is not 8x^5 (or 10x^5, which is what I think you probably meant to write) by trying a value or two for x. For example if x=2 then 4x^2 + 6x^3 = 4(2)^2 + 6(2)^3 = 64, which does not equal 8(2)^5 = 256. Asking why they aren't equal is like asking why 2+2 isn't 17. One thought that might help is to think of the exponent as defining a dimension, so that x^3 is equivalent to a 3-dimensional cube and x^2 is equivalent to a 2-dimensional square. You can add cubes together - for example 3 cubes plus 4 cubes = 7 cubes, so 3x^3 + 4x^3 = 7x^3. Or you can add squares together (3 squares + 4 squares = 7 squares), but you can't add squares to cubes.
what you are asking, is:
why isn't x^{a} + x^{b} = x^{a+b}?
well, let's look at some specific values for a, and b. let's pick a = 1, and b = 2 (i like small numbers).
so, we want to see why:
x + x^{2} ≠ x^{3}.
now we could just "plug in" various values for x, and see what happens. but since we're doing algebra, let's let the rules of algebra do the heavy lifting (never compute, if you can avoid it, i always say).
let's play "devil's advocate" and pretend that the two sides ARE equal (like we hope they are).
recall that x^{2} = x*x, and x^{3} = x*x*x.
what we're going to do, is "factor an x out of each side":
x(1 + x) = ? x(x^{2})
with me so far? now, if x = 0, it certainly is true that 0 + 0 = 0, but that's kind of...boring...so let's look at any other x. and hey! if x isn't 0, we can "divide it out":
1 + x = x^{2}
we can also subtract x from both sides, that's always fair:
1 = x^{2} - x.
let's "factor out another x" on the right:
1 = x(x - 1).
hmm. can you see why x can't be an integer? if it was, then x - 1 would also be an integer, so x(x-1) would have to be either 1*1, or -1*-1, and x and x-1 can't be the same integer.
but if it doesn't work for integers, then how can it possibly work "for all x"?
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we could go the other way, and suppose "x" is a specific number, like 2:
could it be that:
2^{a} + 2^{b} = 2^{a+b}, for any a and b? again, we let a = 1, b = 2:
2^{1} + 2^{2} = ? 2^{3}
2 + 4 =? 8
6 =? 8, no, it doesn't look like that works.
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naively, you can think of polynomials as being like integers, only "in base x", instead of "base 10". we don't add the tens place to the hundreds place, so why should we lump x^{2} in with the x^{3}?