Express g(x) as a product of linear factors

Question

G(x)=x^3-3x^2-14x+12

Answer 1

x³-3x²-14x+12

First, what you need to do is to multiply the coefficient of the term with the highest degree with the constant term. (1)(12) = 12

Second, get all the factors of the result and put them in order like this:

±1, ±2, ±3, ±4, ±6, ±12

Third, use each one as a divisor for synthetic division until you get a remainder of zero. Start with the numbers at the middle of the ordered factors (±3, ±4):

3 ¦ 1 -3. -14. 12

______ 3. 0. -42

------------------------

___1. 0. -14. -30

-3 ¦ 1 -3. -14. 12

______ -3. 18. -12

------------------------

_____1 -6. 4. 0

(I stop here because f(-3) = 0)

Using the coefficients, the first 2 factors are [x-(-3)](x²-6x+4) or (x+3)(x²-6x+4).

x+3 is linear but x²-6x+4 is quadratic. To make the quadratic into 2 linear factors we can use of quadratic formula:

x= [6± √(36 - 16)]/2

x=(6± √20)/2

x=(6± 2√5)/2

x=3± √5

Therefore the three linear factors of x³-3x²-14x+12 are:

(x+3)[x-(3+√5)][x-(3-√5)] or

G(x)= (x+3) (x-3-√5) (x-3+√5)