(x²-x+1)2- (x + 1)³ + 3(x − 1)(x² + x + 1) = x(x³-5) -3
#Prove sameness
kind garnet
drowsy hingeBOT
kind garnet
Help?
kind garnet
Expand both sides.
Yes
kind garnet
I don't know, it has to be solved
stuck holly
Did you expand both sides?
kind garnet
Yes
stuck holly
What were your results for that?
kind garnet
Can you solve it?
kind garnet
What were your results for that?
Moment
stuck holly
Yes.
stuck holly
Is that what you got from the left side?
stuck holly
OK, so expand the left side.
kind garnet
I can't, I don't know how
stuck holly
OK, expand (x^2 - x + 1)^2.
kind garnet
Can you give me a solution, I will look at it and understand it
Because it's a little hard for me like that
stuck holly
The solution is that you expand both sides, then simplify, then see that they're equal.
In order to expand the left side, you just expand the parts of it.
kind garnet
The solution is that you expand both sides, then simplify, then see that they're...
I know, but I don't know how to solve it
stuck holly
That is a proof that they're equal.
kind garnet
Can you just solve it for me?
stuck holly
No, we can only help you to know how to do it.
kind garnet
ok
stuck holly
The way you prove that two things are equal is to do correct algebra steps to get them to look equal.
kind garnet
I do not know how
stuck holly
You have (x² - x + 1)² - (x + 1)³ + 3(x − 1)(x² + x + 1).
So, break it into three parts.
Part 1 is (x² - x + 1)². Part 2 is (x + 1)³. Part 3 is (x − 1)(x² + x + 1).
Can you expand those three parts separately?
Like, what is (x² - x + 1)² after you expand it?
kind garnet
Well, I can't just thank you
stuck holly
OK, so here's how you do that.
kind garnet
for taking the time
stuck holly
Squaring something means to multiply it by itself.
So, you have (x² - x + 1)², and that can be turned into (x² - x + 1)(x² - x + 1).
kind garnet
Yes
stuck holly
Then, you do this: x²(x² - x + 1) - x(x² - x + 1) + 1(x² - x + 1).
Notice how I took the three terms on the left and multiplied them by the right.
Like we had x² - x + 1. The three parts are x², -x, and 1.
kind garnet
👍
stuck holly
And then I wrote each of those parts from the left factor in front of the right factor.
To see it a bit clearer.
(a + b + c)(d + e + f) = a(d + e + f) + b(d + e + f) + c(d + e + f).
Does it make sense how that works?
kind garnet
Yes
stuck holly
OK, so do that with (x² - x + 1)². First notice that squaring means multiplying a number by itself and write it as (x² - x + 1)(x² - x + 1). Then do that trick I just showed you. Then use the distributive property.
kind garnet
Yes after that?
stuck holly
Oh, just do it up to there. Let's see what you get.
Then, we can do the same sort of thing for part 2 and part 3.
kind garnet
That's enough for me thanks
stuck holly
OK.
kind garnet
I'll try to solve it later, if I can't do something, I'll write to you
stuck holly
OK, you can also ping @Helpers 15 minutes after you ask.
kind garnet
OK
kind garnet
@stuck holly
stuck holly
<@537482727557955597>
Hello, I'm back.
For when you get back, first do:
(x² - x + 1)²
(x² - x + 1)(x² - x + 1)
x²(x² - x + 1) - x(x² - x + 1) + 1(x² - x + 1)
(x²x² - x²x + x²) - (xx² - xx + x1) + (1x² - 1x + 1*1)
Then you simplify everything.
For the (x + 1)³ part, you do this:
(x + 1)³
(x + 1)(x + 1)(x + 1)
Parenthesize the first two to do them first:
[(x + 1)(x + 1)](x + 1)
[x(x + 1) + 1(x + 1)](x + 1)
[xx + x1 + 1x + 1 * 1](x + 1)
Then simplify the part in the square brackets. Then do the same thing, multiply each term in the simplified square brackets by the (x + 1), then simplify.
For the 3(x − 1)(x² + x + 1) part, parenthesize the first two:
3(x − 1)(x² + x + 1)
[3(x − 1)](x² + x + 1)
Distribute:
[3x - 3*1](x² + x + 1)
Then, simplify that and do the same sort of thing.
Then, you take the simplified first part, subtract the simplified second part from it, and add the simplified third part. Then simplify.
It's just one small technique done a lot of times.
stuck holly
x⁴-5x-3
When you simplify, you should get this ^
If you don't get that, show your work and ask for help on where you went wrong.
Once you get x⁴ - 5x - 3 on both sides of the equal sign, it's proven because when you do algebra steps the right way to two things and they end up the same, that means they have to be equal.