OK, hopefully you’ve managed to do Ex 1B successfully and practised those well.
If you look at Example 8 on page 9, you’ll see how you can sometimes get a remainder when you finish the division. It’s a pretty simple extension of what you’ve done before. Significance of having a remainder? It means that the thing you’re dividing by (in that example (x-4) is not a factor of 2x^3 – 5x^2 -16x + 10.
Then if you look at Example 7, it’s simply showing an example of when you get a ‘zero’ bit part way through. If that happens, you just bring down the next bit of the polynomial. May be best if I go through that in person.
Factor Theorem:
The Factor Theorem is really simple. It just states that for any polynomial f(x), if f(p) = 0, then (x – p) is a factor of f(x).
Eg, show (x – 2) is a factor of x^3 + x^2 – 4x – 4
f(2) = 2^3 + 2^2 – (4 x 2) – 4 = 0, so (x – 2) is a factor.
You have to learn the basic proof of this, which is in Example 12 on page 12.
Do Ex 1D
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