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Certifictae in further mathematics (factorising quadratics with cubed unknowns)
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Okay so I am studying the aqa certificate in further mathematics and I would like some help. I don't understand how to factorise quadratics which also have cubed unknowns (e.g. x^3 +x^2 +2x +1). I know that they are meant to have three brackets but I have no idea how to figure out what should be in the brackets. My class has a test on Wednesday January 6th and I am worried that this might come up on the test. I've been looking online and I haven't found anything that explains how to do it so I would very appreciative if someone could help me.
Thanks for reading and more thanks if you can help.
Thanks for reading and more thanks if you can help.
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#2
(Original post by AspiringUnderdog)
Okay so I am studying the aqa certificate in further mathematics and I would like some help. I don't understand how to factorise quadratics which also have cubed unknowns (e.g. x^3 +x^2 +2x +1). I know that they are meant to have three brackets but I have no idea how to figure out what should be in the brackets. My class has a test on Wednesday January 6th and I am worried that this might come up on the test. I've been looking online and I haven't found anything that explains how to do it so I would very appreciative if someone could help me.
Thanks for reading and more thanks if you can help.
Okay so I am studying the aqa certificate in further mathematics and I would like some help. I don't understand how to factorise quadratics which also have cubed unknowns (e.g. x^3 +x^2 +2x +1). I know that they are meant to have three brackets but I have no idea how to figure out what should be in the brackets. My class has a test on Wednesday January 6th and I am worried that this might come up on the test. I've been looking online and I haven't found anything that explains how to do it so I would very appreciative if someone could help me.
Thanks for reading and more thanks if you can help.
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#3
(Original post by AspiringUnderdog)
Okay so I am studying the aqa certificate in further mathematics and I would like some help. I don't understand how to factorise quadratics which also have cubed unknowns (e.g. x^3 +x^2 +2x +1). I know that they are meant to have three brackets but I have no idea how to figure out what should be in the brackets. My class has a test on Wednesday January 6th and I am worried that this might come up on the test. I've been looking online and I haven't found anything that explains how to do it so I would very appreciative if someone could help me.
Thanks for reading and more thanks if you can help.
Okay so I am studying the aqa certificate in further mathematics and I would like some help. I don't understand how to factorise quadratics which also have cubed unknowns (e.g. x^3 +x^2 +2x +1). I know that they are meant to have three brackets but I have no idea how to figure out what should be in the brackets. My class has a test on Wednesday January 6th and I am worried that this might come up on the test. I've been looking online and I haven't found anything that explains how to do it so I would very appreciative if someone could help me.
Thanks for reading and more thanks if you can help.
The example you've provided isn't factorisable, so if we take something that is, eg x^3 + x^2 - x - 1.
The factor theorem tells us that if, some number a, is a root (i.e f(a) = 0) then it can be written as f(x) = (x-a)(q(x)) where q(x) is some other function, in this case a quadratic, that you have to find and solve.
Eg if 5 was a root of that equation, then x^3 + x^2 - x - 1 = (x-5)(q(x)). You then have to use other things that you know to find q(x). Hint: (x^3 + x^2 - x - 1)/(x-5) = q(x). But 5 isn't necessarily a root.
And how do you find a root? Well, see if you can spot it from the equation.
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#4
(Original post by Duke Glacia)
theres no exact way to factorise a cubic equation(unlike quadratic eqn ). so try numbers like x=1,2,3,5.-1,-2,-3,-5,-7 and use factor theorum to factorise
theres no exact way to factorise a cubic equation(unlike quadratic eqn ). so try numbers like x=1,2,3,5.-1,-2,-3,-5,-7 and use factor theorum to factorise
But none (in radicals) for 5 order and above IIRC
https://en.wikipedia.org/wiki/Galois...uintic_example
Edit: OP, do NOT try to remember this formula!
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#6
For the quintic and beyond, one should note that they are solvable in terms of special functions called modular functions. Alas, this fascinating theory is rarely taught even at undergraduate level.
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#7
(Original post by Gregorius)
You might find the Wikipedia articles on the cubic and on the quartic a bit more illuminating, as they unwrap the process of solving such equations by radicals. Beyond the quartic, whether a given polynomial equation is solvable by radicals depends on whether a particular group (that basically arises as a permutation group of the roots of the equation) is solvable.
For the quintic and beyond, one should note that they are solvable in terms of special functions called modular functions. Alas, this fascinating theory is rarely taught even at undergraduate level.
You might find the Wikipedia articles on the cubic and on the quartic a bit more illuminating, as they unwrap the process of solving such equations by radicals. Beyond the quartic, whether a given polynomial equation is solvable by radicals depends on whether a particular group (that basically arises as a permutation group of the roots of the equation) is solvable.
For the quintic and beyond, one should note that they are solvable in terms of special functions called modular functions. Alas, this fascinating theory is rarely taught even at undergraduate level.
Aren't modular functions quite new? IIRC, they were involved in Wiles's proof of Fermat's Last Theorm
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#8
(Original post by Johann von Gauss)
Can't wait to learn Galois Theory
Aren't modular functions quite new? IIRC, they were involved in Wiles's proof of Fermat's Last Theorm
Can't wait to learn Galois Theory
Aren't modular functions quite new? IIRC, they were involved in Wiles's proof of Fermat's Last Theorm
Modular forms and functions are most definitely from the golden age of nineteenth century mathematics; if you pop down to the history section of the Wikipedia article, you'll see that Wiles's work lies very much in the "great tradition" that grew out of Euler, Gauss, Jacobi...
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#10
The reason I'm pointing this out is because I think it's important to bear in mind when using the factor theorem so you know which numbers to try. For GCSE though they will give you 'nice cubics, with 3 rational (real roots). I probably haven't explained this too well but it may be worth looking at the Wikipedia page on the rational roots theorem.
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