Factor, Remainder Theorems and Algebraic Long Division – AS Maths Revision – Core Mathematics (C2)

Factor Stuff

polynomial long division, example 2, showing a...

polynomial long division, example 2, showing a “gap” (i.e., zero coefficient) (Photo credit: Wikipedia)

  • You can simplify algebraic fractions by cancelling common factors
  • To get common factors, you may have to do some factorisation
  • Dividing by a fraction is the same as multiplying by the reciprocal of that fraction (flipping it up-side-down and multiplying)
  • Any number is equal to that number divided by 1 (so everything is a fraction)
  • The factor theorem: if f(x) is a polynomial, and f(a) = 0, you know that (x-a) is a factor of f(x) [and it works backwards, too]
  • The remainder theorem: if polynomial f(x) is divided by (x-a), the remainder is f(a)
  • Also, if f(x) is divided by (bx-a), the remainder is  f(a/b)
  • To completely factorise a cubic polynomial thing, divide it by a known factor, using algebraic long division, then you’re left with a quadratic which you can then break down into the other two factors [there will be three: (x+5)(x-3)(2x+7), for example]
  • I shan’t explain how to factorise quadratics and linear things – that’s GCSE (or maybe AS Core 1)

Algebraic Long Division

  • Draw the bus stop shape thing (I really can’t think how to describe it. It’s a bit like a square root sign)
  • Put the polynomial you’re dividing inside the bus stop
  • Put the divisor (thing you’re dividing by) to the left
  • The quotient (the answer) will be gradually built up along the top of the bus stop, so leave a bit of space above
  • I’m assuming your divisor has two terms: x and something else
  • I’m also assuming your polynomial goes something x cubed, something x squared, something x, something
  • If you don’t have a “something x squared”, for instance, put zero x squared in to hold its place
  • Be very careful with negatives: 1 – – 1 = 2, and – 1 – – 1 = 0
  • The following overly descriptive instructions should be a foolproof way to divide your polynomial by your divisor… but I probably haven’t made it clear…
  1. Divide the first term of the polynomial by x (the first term of the divisor)
  2. Put the result on top of the bus stop, above the first term of the polynomial
  3. Multiply the thing you just wrote on top of the bus stop by the divisor: you should get two terms – one identical to the first term of the polynomial, and another with the same index of x as the second term of the polynomial
  4. Write these terms underneath the first two terms of the polynomial
  5. Draw a horizontal line underneath the two terms you just wrote, and subtract them from the two polynomial terms above them
  6. There should be only one term in the result
  7. Divide this term by x (the first term of the divisor)
  8. Put the result on top of the bus stop, above the second term of the polynomial
  9. Multiply the thing you just wrote on top of the bus stop by the divisor: you should get two terms – one identical to the result of your first subtraction, and another with the same index of x as the third term of the polynomial
  10. Write these two terms underneath the second and third terms of the polynomial (the two identical terms should be directly above/below each other)
  11. Write the third term of the polynomial next to the result of your first subtraction (also draw an arrow from the third term to where you wrote it, to indicate that you’re ‘bringing it down’)
  12. Draw another horizontal line and subtract again (it should be clear what to subtract from what!)
  13. Once again, your subtraction result should be only one term
  14. Divide this term by x (the first term of the divisor)
  15. Put the result on top of the bus stop, above the third term of the polynomial
  16. Multiply the thing you just wrote on top of the bus stop by the divisor: you should get two terms – one identical to the result of your second subtraction, and another which is identical to the final (fourth) term of the polynomial
  17. Write these two terms underneath the third and fourth terms of the polynomial (just like you’ve already done a few times…)
  18. ‘Bring down’ the fourth/final term of the polynomial
  19. Draw the horizontal line and subtract
  20. If the result of this final subtraction is zero, the divisor is a factor of the polynomial. If it’s not zero, either there’s a remainder, as it isn’t a factor (check using the factor and remainder theorems), or you’ve made a mistake…
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About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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