Is there is a shortcut of dividing polynomials?

Answer 1

Yes, first you find the p(x) = 0. I'll give you an example.

x^3-13x+12

p(-4)=-64 + 52 + 12 = 0

so (x+4) is a factor

Now open the bracket for x+4 and fix the other bracket

Firstly, we know that to get x^3 with x, we need to multiply it by x^2

(x+4)(x^2....)

Now, as you can see, by getting the x^3, you have also created a 4x^2. And if you look back into your equation, you need a -13x, so we need to somehow get rid of the 4x^2. We can do this by subtracting 4x from the next bracket.

(x+4)(x^2-4x....)

See how the x will multiply with the -4x to give you -4x^2? But now we have also created a -16x (the 4 x -4x), and the equation wants -13x, so we need to add 3x back.

(x+4)(x^2-4x+3)

Now we have our -13x, but we have also created a +12. Looking back into the rule, we need the +12, and so we have found 2 factors of the polynomial.

Finally, we can simplify the second bracket into another 2 factors, which gives us:

(x+4)(x-3)(x-1) as our factors.