How to use synthetic division and the factor theorem to determine if x - c is a factor of f(x).

for f(x)=3x^3+8x^2-8x-20; x+3

Is x+3 a factor of f(x)?

-3 |... 3 ... 8 .. -8 .. -20
....|........ -9 ... 3 ... 15
....-------------------------------
........ 3 .. -1 ... -5 .. -5

No, (x + 3) is not a factor of 3x³ + 8x² - 8x - 20 because it results in a remainder (the -5 is not a 0).

Well, I suppose you could use synthic division to divide (x+3) into f(x). If the remainder is 0, then
its a factor.

But... that's an awful lot of trouble if you only need to know if it is a factor.

I can think of two ways to tell whether it's a factor.

First, if (x+3) is a factor, then x = -3 must be a root. Compute f(-3). If it isn't 0, then
(x+3) isn't a factor.

3(-3)^3 + 8(-3)^2 -8(-3) - 20 = -81 + 72 + 24 - 20 = -5, so that rules it out.

Even easier, the rational root theorem tells you that any rational root of a polynomial will
be of the form

(some +/- integer factor of the constant term)/(some +/- integer factor of the coefficient of the first term)

If this case, since 3 isn't a factor of 20, there's no way that -3 can be a root.

first of all the factor theorem basically states that, when f(x) is divided by,lets say (x-a) and f(a) = 0 then (x-a) is a factor of f(x)

so if (x-c) is a factor of f(x) then f(c)=0, that is when all x terms in f(x) is replaced by c if the sum of the terms equal to zero ,(x-c) is a factor of f(x)

f(x)=3x^3+8x^2-8x-20
consider f(-3) =3(-3)^3 + 8(-3)^2 -8(-3) -20 = -81 +72 + 24 -20 =not equal to zero
that means x+3 is NOT a factor of f(x)!



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using synthetic division divide f(x) by (x+3) ,again if the remainder is zero ,then it is a factor;

-3 / 3 8 8 -20
3 -1 1-5 -5
i wish i could explain how to perform the division ,but i dont know how to , so hopefully you know the basics