factorise completely f(x) x^3 + 3x^2 - 4x -12?

Question

could someone please help me do this as i dont really have a clue of how to do it! thanks
charlie

Answer 1

Start by guessing at a single factor that will make this zero. Possible factors will be factors of the last coefficient (-12) divided by factors of the first coefficient (1). In other words {±1, ±2, ±3, ±4, ±6, ±12}

With a little trial and error, it looks like x = 2 will make this be zero:

f(2) = 2^3 + 3(2)² - 4(2) - 12 = 8 + 12 - 8 - 12 = 0

So one factor should be (x - 2). Now use synthetic divsion to factor further.
...................... x² +5x + 6
x + 2 ) x^3 +3x² - 4x - 12
............ x^3 - 2x²
............ ------------------
..................... 5x² - 4x
..................... 5x² - 10x
..................... ------------------
..................... ......... 6x - 12
..................... ......... 6x - 12
..................... ......... ----------
..................... ......... ........ 0

f(x) = (x-2)(x² + 5x + 6)

Then you can easily factor the last part:

f(x) = (x - 2)(x + 2)(x + 3)

Answer 2

f(x) = x^3 + 3x^2 -4x - 12
= (x+3)(x^2 - 4)
= (x+3)(x+2)(x-2)

By looking at the equation, you can see that the first two terms contain x^2, and that the coefficients of the last two terms are the same as the coefficients of the first two terms each times (-4). Inspection of the specific terms involved can often lead to seeing that they are multiples of one another.

Answer 3

f(x) = x^3 + 3x^2 - 4x -12
f(x) = x^2(x + 3) - 4(x + 3)
f(x) = (x + 3)(x^2 - 4)
f(x) = (x + 3)(x + 2)(x - 2)

Answer 4

f(x) = x^3 + 3x^2 - 4x - 12

by observation x = 2 is a soltion of f(x)

f(2) = 2^3 + 3(2)^2 - 4(2) - 12 = 8 +12 - 8 - 12 = 0

so (x-2) is a factor.

now divide f(x) by x - 2

x-2)x^3+3x^2-4x- 12(x^2
___x^3-2x^2
_______________
_____+5x^2-4x(5x
_____+5x^2-10x(
______________
__________+6x-12(6
__________+6x-12
________________
_____________0

So the quotient is x^2 + 5x + 6

factorizing the quotient

x^2 + 5x + 6 = (x+2)(x+3)

f(x) = (x-2)(x+2)(x+3)

Answer 5

x^3 + 3x^2 - 4x - 12
x^2(x+3)-4(x+3)
(x^2 -4)(x+3)
(x-2)(x+2)(x+3)

Answer 6

for x=-3 f(x)=0, so (x+3) is a factor.
divide f(x) by (x+3), you get (x^2-4)=(x+2)(x-2)
so (x+3)(x+2)(x-2) is the answer.
-----x^2-4---------------
x+3|x^3+3x^2-4x-12
----|x^3+3x^2
continue

Answer 7

do a smart guess : try divisors of 12 ; 1, 2,3,4,and 6