STEP Formulae/ Identities

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  1. brittanna's Avatar
    1 14-06-2012  22:59
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    STEP Formulae/ Identities
    I was just wondering what formulae and identities you need to remember (aren't given in formulae booklet) for STEP (1, 2 AND 3).

    This is because I was doing a question that involved using the identity for the difference of two cubes which I hadn't come across. So are there any other identities like this that you need to know about?
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  2. raheem94's Avatar
    2 14-06-2012  23:42
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    Re: STEP Formulae/ Identities
    (Original post by brittanna)
    I was just wondering what formulae and identities you need to remember (aren't given in formulae booklet) for STEP (1, 2 AND 3).

    This is because I was doing a question that involved using the identity for the difference of two cubes which I hadn't come across. So are there any other identities like this that you need to know about?
    Which identity are you referring to?

    Are you talking about this one p^3 - q^3 = (p-q)(p^2+pq+q^2) \ ?
  3. brittanna's Avatar
    3 14-06-2012  23:54
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    Re: STEP Formulae/ Identities
    (Original post by raheem94)
    Which identity are you referring to?

    Are you talking about this one p^3 - q^3 = (p-q)(p^2+pq+q^2) \ ?
    Yeah.

    I haven't finished A level maths yet, but it doesn't seem like something you would learn in C3 or C4.
  4. Stray's Avatar
    4 14-06-2012  23:55
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    Re: STEP Formulae/ Identities
    If you like that one, you'll also like its companion...

    p^3 + q^3 = (p+q)(p^2 - pq + q^2)
    Last edited by Stray; 15-06-2012 at 00:07. Reason: DUH - too late at night
  5. raheem94's Avatar
    5 14-06-2012  23:59
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    Re: STEP Formulae/ Identities
    (Original post by brittanna)
    Yeah.

    I haven't finished A level maths yet, but it doesn't seem like something you would learn in C3 or C4.
    It isn't in C3/C4 but i learned this in year 8 so it is probably an extremely common identity.
  6. raheem94's Avatar
    6 15-06-2012  00:03
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    Re: STEP Formulae/ Identities
    (Original post by SubAtomic)
    How is that true:confused:
    Its a typo, it should have been, p^3 + q^3 = (p+q)(p^2 - pq + q^2)
  7. SubAtomic's Avatar
    7 15-06-2012  00:04
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    Re: STEP Formulae/ Identities
    (Original post by raheem94)
    Its a typo, it should have been, p^3 + q^3 = (p+q)(p^2 - pq + q^2)
    Funny typo considering p and q are at opposite ends of a keyboard.
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  8. brittanna's Avatar
    8 15-06-2012  00:04
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    Re: STEP Formulae/ Identities
    I had only ever learned the difference of two squares. What other 'extremely common identities' are there that I need to know about that are quite often needed to answer questions?
  9. Stray's Avatar
    9 15-06-2012  00:06
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    Re: STEP Formulae/ Identities
    (Original post by SubAtomic)
    How is that true:confused:
    Good point... I had mistyped it. Sorry 'bout that. Yes - not so much a typo as a brain-fail.
    Last edited by Stray; 15-06-2012 at 00:08.
  10. raheem94's Avatar
    10 15-06-2012  00:06
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    Re: STEP Formulae/ Identities
    (Original post by SubAtomic)
    Funny typo considering p and q are at opposite ends of a keyboard.
    lol

    (Original post by brittanna)
    I had only ever learned the difference of two squares. What other 'extremely common identities' are there that I need to know about that are quite often needed to answer questions?
    I just said extremely common because it was year 8 stuff.

    I don't know which ones you are expected to know.
  11. Slumpy's Avatar
    11 15-06-2012  00:06
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    Re: STEP Formulae/ Identities
    (Original post by brittanna)
    Yeah.

    I haven't finished A level maths yet, but it doesn't seem like something you would learn in C3 or C4.
    I wouldn't really call it an identity, so much as something you can just spot. If you know the factor theorem (pretty sure this is early A level), it's easy to come up with also.
  12. SubAtomic's Avatar
    12 15-06-2012  00:45
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    Re: STEP Formulae/ Identities
    (Original post by Slumpy)
    I wouldn't really call it an identity, so much as something you can just spot. If you know the factor theorem (pretty sure this is early A level), it's easy to come up with also.
    Yep, seen these cubics and some other various equations just before a section on roots, polynomials and factors/factorisation on a pdf, will cut and paste a bit(no I wont I will have to tex it because it won't bloody accept the paste in tex:rolleyes:).

    \displaystyle 1) \ x+xy=x(1+y) \ , \mathrm{factors} \ x \ \mathrm{and} \ 1+y. \\ \\ 2) \ x^2+x-2=(x-1)(x+2). \\ \\ 3) \ x^3-y^3=(x-y)(x^2+xy+y^2). \\ \\ 4) \ x^3+y^3=(x+y)(x^2-xy+y^2). \\ \\ 5) \ x^3+3x^2+3x+1=(x+1)^3 . \\ \\ 6) \ a^4-b^4=(a^2-b^2)(a^2+b^2)=(a-b)(a+b)(a^2+b^2). \\ \\ \mathrm{You\ can\ work\ this\ out\ from\ rewriting} \ a^4-b^4=(a^2)^2-(b^2)^2 \\ \mathrm{and\ then\ use\ the\ difference\ of\ two\ squares\ factorisation}. \\ \\ 7) \ x^4+y^4=(x^2+ \sqrt 2 xy+y^2)(x^2- \sqrt 2 xy +y^2). \\ \\ (\mathrm{not\ expected\ to\ know\ this})
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  13. ian.slater's Avatar
    13 15-06-2012  07:33
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    Re: STEP Formulae/ Identities
    (Original post by brittanna)
    I was just wondering what formulae and identities you need to remember (aren't given in formulae booklet) for STEP (1, 2 AND 3).

    This is because I was doing a question that involved using the identity for the difference of two cubes which I hadn't come across. So are there any other identities like this that you need to know about?
    It's worth knowing the general solutions to:

    sin(x) = sin(y)

    and the cos and tan equivalents.

    The formula book does contain the identities for things like:

    cosAcosB = ...

    but it's worth being aware of them as they are often not explicitly taught.
  14. ben-smith's Avatar
    14 15-06-2012  07:53
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    Re: STEP Formulae/ Identities
    Tanx/2=\dfrac{1-cosx}{sinx} Tanx=Cot(\pi/2-x)
    You'd be surprised how often these come up.
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  15. SubAtomic's Avatar
    15 15-06-2012  17:13
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    Re: STEP Formulae/ Identities
    Don't know if anything I am posting is of any use as I aren't doing STEP. So correct me or ignore me

    Here is another equation I have been introduced to today.

    \displaystyle x^2-x-1=0 \ \ ,\mathrm{aka\ the\ golden\ ratio\ equation\ ,\ it\ has\ solutions} \\ \\ \ x=\frac{1}{2}(1+ \sqrt 5)=1.6180...=\phi \ \mathrm{and} \ x=\frac{1}{2}(1- \sqrt 5)=-0.6180...=\psi


    \displaystyle \phi+\psi=\frac{1}{2}(1+\sqrt 5)+\frac{1}{2}(1-\sqrt 5) \\ \\ =\frac{1}{2}+\frac{1}{2}=1

    And some stuff about using exact forms of \displaystyle \phi \ \mathrm{and} \ \psi to verify properties of \displaystyle \phi \ \mathrm{and} \ \psi

    For instance

    \displaystyle \phi \psi=-1\ \ \mathrm{and} \ \phi-\psi=\sqrt 5
    Last edited by SubAtomic; 15-06-2012 at 17:33.
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