how do u factor x^3+3x^2-12x+4?

Answer 1

x^3+3x^2-12x+4
The Factor Theorem says a polynomial p(x) has x-b as a factor if, and only if, p(b)=0
Translation: Pull any number out of the air. Replace x in your expression with that number. If the expression
works out to have a value of zero, then (x- that number)
is a factor.
In your problem, picking 2 gives 8+12-24+4. This adds up to 0, so (x-2) is a factor.
You now divide x^3+3x^2-12x+4 by (x-2), and you will get (x-2)(x^2+5x-2).
I can't easily factor (x^2+5x-2), so I have 2 choices.
Choice 1, Leave it as is.
Choice 2. Use the quadratic formula. Doing this gives
2 very messy answers.
Stick with Choice 1. Answer is (x-2)(x^2+5x-2)

Answer 2

anything over ^2 use long division. notice the factors of the constant term (+4). 1*4, 2*2, -1*-4, -2*-2

since there is a two digit coefficient on one of the terms,
try (x + 4)
x^2(x + 4) = x^3 + 4x^2
x^3 + 3x^2 - 12x + 4 - (x^3 + 4x^2) = -x^2 - 12x + 4
-x(x + 4) = -x^2 - 4x
-x^2 - 12x + 4 - (-x^2 - 4x) = -8x + 4
nothing multiplied by (x + 4) will give you -8x + 4
so try (x - 4)
x^2(x - 4) = x^3 - 4x^2
x^3 + 3x^2 - 12x + 4 - (x^3 - 4x^2) = 7x^2 - 12x + 4
7x(x - 4) = 7x^2 - 28x
7x^2 - 12x + 4 - (7x^2 - 28x) = 16x + 4
nothing multiplied by (x - 4) will give you 16x + 4
so that leaves 2 and -2. since there is a negative term
try (x - 2)
x^2(x - 2) = x^3 - 2x^2
x^3 + 3x^2 - 12x + 4 - (x^3 - 2x^2) = 5x^2 - 12x + 4
5x(x - 2) = 5x^2 - 10x
5x^2 - 12x + 4 - (5x^2 - 10x) = -2x + 4
-2(x - 2) = -2x + 4
-2x + 4 - (-2x + 4) = 0 that's it
the factors so far are (x - 2)(x^2 + 5x - 2)
can x^2 + 5x - 2 be factored?
does any combination of factors of -2 add up to +5?
factors of -2: 1*-2 and _1*2, so the answer is no
you would have to complete the square
x^2 + 5x + 25/4 - 2 - 25/4 = (x^2 + 5x + (5/2)^2) - 8/4 - 25/4 =
(x + 5/2)^2 - 33/4 we can use the difference of squares here
(x + 5/2)^2 - 33/4 = (x + 5/2)^2 - (sqrt33/2)^2 =
(x + 5/2 + sqrt33/2)(x + 5/2 - sqrt33/2) =
(x + (5 + sqrt33)/2)(x + (5 - sqrt33)/2)
so the three factors of x^3+3x^2-12x+4 are
(x - 2)(x + (5 + sqrt33)/2)(x + (5 - sqrt33)/2)

Answer 3

(x-2)(x^2 + 5x - 2)

I got this by using synthetic division with x=2.