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# Factor Theorems Question watch

1. (Original post by Retsek)

Try expanding the brackets and comparing coefficients In particular, what can you tell about the constant term 150 in terms of and ?
2. (Original post by crashMATHS)
Try expanding the brackets and comparing coefficients In particular, what can you tell about the constant term 150 in terms of and ?
And then does it become a weird sort of simultaneous equation?

d(c^2) = 150
3. Equate terms in from of the x terms on the original equation and your expanded brackets.
4. (Original post by Anfanny)
Equate terms in from of the x terms on the original equation and your expanded brackets.
This feels really messy and not right

Have I expanded the brackets correctly?

(x+c)^2 (x+d) = x^3 + x^2(2c + d) + x(2dc + c^2) + dc^2
5. (Original post by Retsek)
And then does it become a weird sort of simultaneous equation?

d(c^2) = 150
Essentially, yeah. You can expand the bracket and equate the coefficients and solve the system of equations.

However, you do also have quite a nice thing going on here. Notice that c is squared (and is an integer). So you need a square number * some other number = 150. That should make it clear what c and d have to be. Then substitute those in and expand to find a and b.
6. (Original post by crashMATHS)
Essentially, yeah. You can expand the bracket and equate the coefficients and solve the system of equations.

However, you do also have quite a nice thing going on here. Notice that c is squared (and is an integer). So you need a square number * some other number = 150. That should make it clear what c and d have to be. Then substitute those in and expand to find a and b.
So there's only one possible solution for
d(c^2) = 150
if c is an integer?

Also I got these but idk if they're leading anywhere
2c + d = a
2dc + c^2 = b
There's too many unknowns right?

EDIT: I think I've got it
7. (Original post by Retsek)
d(c^2) = 150
if c is an integer?
The key thing is that c^2 is a square number. So you can easily find out what c^2 is by thinking about what square numbers divide 150
8. (Original post by crashMATHS)
The key thing is that c^2 is a square number. So you can easily find out what c^2 is by thinking about what square numbers divide 150
a = 16
b = 85
c = 5
d = 6

Thanks so much for your help!
9. (Original post by Retsek)
Expanding out and equating everything, you should get this equation

By inspection, if and were to be positive integers positive integer. The only numbers that divide cleanly into is . Since, is a positive integer that needs to be squared. , since means integer. means integer. Continuing this argument, we arrive at that , since which is a positive integer. d is now easy to to find and so is the other letters.

I think something can be done with the first and second derivative, you might want to look at that
10. (Original post by Retsek)
This feels really messy and not right

Have I expanded the brackets correctly?

(x+c)^2 (x+d) = x^3 + x^2(2c + d) + x(2dc + c^2) + dc^2
It should not feel messy. The x terms are separated from the constants this looks lovely and neat.

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