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Sergeant Loi’s Mid-Year Boot Camp 2008 – Finding Your Roots With Remainder & Factor Theorems

Tuition given in the topic of A-Maths Tuition Notes & Tips from the desk of Sergeant Loi at 5:30 pm (Singapore time)
Sergeant Loi's Helmet

This heavy thing is causing permanent damage
to her previously-rebonded hair

One more time …

*Puts on helmet*


*Sounds bugle* EVERYBODY FALL IN!

Most of you would’ve started your Mid-Year Exams by now – a series of no-holds-barred trials to determine once and for all if indeed a wind tunnel exists between your ears to test your understanding of topics taught in Semester 1.

For some, this could also be a time for your ‘chers to avenge all the tortures you’ve subjected them to throughout the term. As such, the Mids always tend to be a little on the sadistic side, and strewn with devious tricks around every turn and corner.

That’s why Miss Sergeant Loi (whose Teresa Teng voice is now hoarse from all the shouting) is here – to hopefully help save you a mark or two, to give you that little bit of edge from being pwned by your ‘chers.

So let’s do this one more time (to complete the chapter on Polynomials) … for now

Remainder & Factor Theorems


If f(x) is divided by (x a) ⇒ the remainder is f(a)

e.g. Find the remainder when 4x3 – 5x + 1 is divided by:
i. x-2, ii. x+3, iii. 2x-1

Ans: Let f(x) = 4x3 – 5x + 1. Remainder, R =

  1. f(2) = 4(2)3-5(2)+1 = 23
  2. f(-3) = 4(-3)3-5(-3)+1 = -92
    → note it’s divided by x (+) 3 so you’re have to sub in (-)3 instead
  3. f(½) = 4(½)3-5(½)+1 = -1
    → note when divided by (2x-1) → you’ll have to convert it to the form (x-½) first and then sub in the ½

Warning! DON’T waste time doing long division in remainder theorem questions!!!


If f(x) is divided by (xa) and the remainder is 0 ⇔ f(a)=0
⇒ (xa) is a factor of f(x)
⇒ f(x) is exactly divisible by (xa)

From your Sec Two Expansion & Factorisation chapter:

  1. Expansion → remove brackets
  2. Factorisation → put back brackets ⇒ final answer must always be in brackets!

e.g. Factorise x2 – 5x + 6

Ans via Trial & Error (try getting this under 10 sec 😀 ):

Choose 2 factors of the constant 6
try: 1 x 6 → 1x + 6x ≠ -5x (reject)
try: 2 x 3 → 2x + 3x ≠ -5x (reject)
try: (-2) x (-3) → -2x + (-3)x = -5x (YAY!)
⇒ cross-check: f(3) = f(2) = 0 (YAY!)
x2 – 5x + 6 = (x – 3)(x – 2)

Warning! When you see the keyword Factorise, final answer must be in (brackets) i.e. don’t try to be funny and write x = 3, 2 → minus marks!


When you spot the keywords Solve and/or = 0 in your exam question, it means you’ll normally have to:

  1. Find the factors of an equation f(x) (usually cubic)
  2. Find the roots of f(x)=0 (i.e. final answer must be in the form: x = a, b … where a, b, … are the roots)

e.g. Solve 3x3 – 10x2 + 9x – 2 = 0

Ans: Let f(x) = 3x3 – 10x2 + 9x – 2.

  1. Find the first factor via trial and error

    Try x=1: f(1) = 3(1)3 – 10(1)2 + 9(1) – 2 = 0 (YAY!)
    ⇒ (x-1) is a factor

  2. Find the remainder expression by either COMPARING COEFFICIENTS:

    Factorization by Comparing Coefficients

    OR LONG DIVISION (if you’re a long division aficionado)

    Factorization by Long Division

    You should get the SAME expression either way – use which ever method you’re more comfortable with (use one method to cross check the other if you’re one of those with lotsa free time left in your exam).

  3. Factorize the remaining quadratic expression 3x2-7x+2 (via quick Trial and Error method described in B above):

    Choose 2 factors of the constant 2
    try: (-2) x (-1) → (3)(-2)x + (-1)x = -7x (YAY!)
    → Note the coefficient of 3 of the x2 term
    ⇒ 3x2 – 7x + 2 = (3x – 1)(x – 2)
    → Note it’s NOT (3x – 2)(x – 1) coz you need to corss-multiply
    ⇒ cross-check: f(⅓) = f(2) = 0 (YAY!)

    ⇒ 3x3 – 10x2 + 9x – 2 = (x-1)(3x-1)(x-2) = 0
    x = 1, 2, ⅓

Warning! When you see the keywords Solve and/or = 0, final answer must be in the form x = a, b … i.e. don’t stop at factorising → minus marks!
Warning! Sometimes the quadratic equation in Step 3 cannot be easily factorised → you’ll have to use the Quadratic Formula to find the two solutions. You’ll normally get the hint when you see terms like ±√ within the question.


The cubic polynomial f(x) is such that the coefficient of x3 is -1 and the roots of the equation f(x) = 0 are 1, 2 and k. Given that f(x) has a remainder of 8 when divided by x-3, find

  1. the value of k,
  2. the remainder when f(x) is divided by x+3


Since 1, 2 and k are roots, a(x-1)(x-2)(xk) = 0
→ straightaway write down in factorized form once roots are known
→ always remember to include the coefficient a for x3 for it may not always be 1!

And since coefficient of x3 = -1
a = -1
⇒ (-1)(x-1)(x-2)(xk) = 0

Let f(x) = (-1)(x-1)(x-2)(xk)

  1. Since remainder is 8 when divided by (x-3),
    f(3) = (-1)(3-1)(3-2)(3-k) = 8
    (using Remainder Theorem from A above)
    k = 7
  2. Now using k = 7 above, f(x) = (-1)(x-1)(x-2)(x7)

    Remainder when divided by x+3:
    → f(-3) = (-1)((-3)-1)((-3)-2)((-3)-7) = 200

*For some reason, students have a habit of expanding the entire expression after they’ve written down everything in factorized form = what a waste of time. Tsk.

As always, get these rules drilled into your head! Spot the pointers and common mistakes in red! Understand the representative sample question! Check out further questions on factor theorem!

Print this out if necessary and remember the above procedures by heart … and do let Miss Loi know which topics and stuffs you would like to see in her next set of Maths Notes 😉

Till then, understand that the ultimate root of your own equation is to prepare youself in mind and in soul for the Great War at year’s end. So don’t be afraid to make all the mistakes you need to make now (as long as you know what mistakes you’re making!).

Good Luck For Your Mids!

P.S. To the reader who longs and yearns to see Miss Loi’s divine face again, a very heartbroken and upset Sergeant Loi was last seen charging out of camp with her Katana sword, vowing to hunt this reader down, and slice him into many pieces and use the Remainder Theorem to turn whatever that remains of him into 人肉叉烧包 (human buns)!

Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

  1. Leave A Comment!
Comments & Reactions


  1. Someone's Avatar
    Someone commented in tuition class


    I think you missed division by general factors like (ax+b).

  2. La papillion's Avatar
    La papillion commented in tuition class


    Hi Miss Loi,

    Haha, I enjoyed your sergent loi's series 🙂 It's very cute and enlightening. Sorry for not visiting your blog these days, busy like mad dog, as you should know 🙂

    Have fun teaching!

  3. tohkiatshengKEN's Avatar
    tohkiatshengKEN commented in tuition class


    siah lah abang !
    today got rugby match meh ?
    SJI v.s ACSI?
    siah lah abang !!
    saya malayu shiol!
    say i stupid mat dunno how to play rugby issit? take ball round all around the shop like one mat lidat then later pergi kolong blok one korner then chillak sup rokok issit?

  4. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    *Stumbles home and feeling high from excessive inhaling of joss sticks smoke*

    Someone: Are you referring to the extended definition of the Factor Theorem?


    (ax - b) is a factor of f(x) ⇔ f(b/a) = 0

    For this case, Miss Loi didn't state this as it's rare in O-Level solution of cubic equation questions where the first factor (see step C1 above) is a fraction i.e. substituting a b/a value into f(x) for your trial and error.

    HOWEVER we can't rule out such a root appearing in the remaining quadratic equation (see step C2 above) but you should be able to find it easily either via the via quick Trial and Error method (described in B above) or the quadratic formula.

  5. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    Welcome back Papilllion! As you can see from the lightning-speed of Miss Loi's reply, we're all mad dogs running wild this week 😀

  6. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    All views expressed by a certain Toh Kiat Sheng Ken in this comments box are solely those of Toh Kiat Sheng Ken (be it sober or otherwise at the time of writing - likely the latter due to probable hangover from birthday activities) and most definitely do not represent of noble views of this blog!

    Happy Birthday! 😀

  7. Someone's Avatar
    Someone commented in tuition class


    Whoops, I meant the remainder theorem as a whole. Haha.

  8. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class



    Following the extended definition of the Factor Theorem in comment #4, we can similarly generalize the Remainder Theorem to this:

    If f(a) is divided by (ax - b) ⇒ the remainder is f(b/a)

    This being last-minute notes for last-minute students, Miss Loi didn't state it this way so as to minimize any confusion with most textbooks (which only state the case for (x-a)) - though you're expected to know how to find the remainder when f(x) is divided by (ax-b)

    That's why Miss Loi has specially included part iii. (as a reminder) in the example question in A. 😉

  9. TYS's Avatar
    TYS commented in tuition class


    Hi Miss Looi, care to share tips on geometrical proofs (A Maths) since it is a relatively new topic and hard to grasp for most ppl. Thks.

  10. Subtraction's Avatar
    Subtraction commented in tuition class


    Synthetic division allows us to find out the divisibility of your function f(x) by your linear expression k(x)=x-a, where \deg\,f\,>\,\deg\,k, remainder and quotient quickly.

    Is this method allowed in A level?

  11. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    Oops missed out the comments here!

    TYS: Believe you've already heard of the fabled The 阴阳眼 Of Plane Geometry?

    Welcome to Jφss Sticks Subtraction!

    Like the The Cover-Up Rule, Synthetic Division isn't officially taught in all schools and are not covered by most textbooks here.

    Technically, it should not be a problem if you run amok and use this method rampantly in your exams, but you'll also run a slight risk of your examiner/marker getting startled or even slightly annoyed (yeah sad but true) by those 'pagan' workings on your script (especially if it's not officially taught at your school).

    So always, always cover yourself by writing a BIG "Using Synthetic Division," ... at the start of your working!

    Those who're interested in this method will find the following video useful:

    While we'll all agree that this is really a handy method especially when used to check your answer, an old-school Miss Loi would still advocate that you make sure you get your foundations right first before running riot with it.

    P.S. This should be less of an issue at A Level (if that's what you're asking) but nevertheless it's always good practice to declare your method before every major chunk of your hieroglyphic workings to make it easier for your marker to follow (after all, marking ain't a relaxing job) - so that you'll still give him/her a reason to award method marks if you've made a careless mistake.

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