Algebra question - Why is it factored this way?

Question

T^3 + 3T - 12T + 4
_______________
T^3 - 4T

The part is factored like (T-2)(T^2 + 5T -2), when multiplied back would give you T^3 + 3T - 12T + 4 (the numerator). Question is: WHY is it (T-2)? What's the logic behind choosing (T-2)?

Answer 1

Example
4 is a factor of 12 because 12 may be divided by 4 with no remainder.
Similarly t³ + 3t - 12t + 4 may be divided by t - 2 so t - 2 is a factor of t³ + 3t - 12t + 4

t-2 is a factor because there is remainder = 0 when t³ + 3t - 12t + 4 is divided by t - 2.

t - 2 is shown to be a factor by using synthetic division:-
"2 |1"""3"""-12""""4
"" |"""""2""""10"""-4
"""|1"""5""""-2""""0
(t - 2) (t² + 5t - 2)

Trial and error is used by trying t - 1 , t + 1 as factors (they are not).
Logically this leads to trying t - 2 and setting up the division as shown above.

Answer 2

There exists in math something called the FACTOR THEOREM. The formal statement goes like this:
A polynomial P(x) has x-b as a factor if and only if P(b)=0
The layman's translation of this is: pull a number out of the air. Replace x with the number you picked. If the expression works out to = 0, then (x- that number) is a factor of the expression.
For your expression (and I've taken the liberty of re-writing 3T as 3T^2), 2, when substituted into it for x, yields a value of 0 for the expression. TaDah!!
(T-2) is a factor
You divide (T-2) into T^3+3T^2-12T+4 and you get
T^2+5T-2.
And that's the logic.

Answer 3

The graph Y = T³ + 3T² - 12T + 4 crosses the x-axis at T = 2. This gives us the factor (T - 2). They chose that factor either using division of polynomials or synthetic division.

Answer 4

If you see the denominator, (T^3 - 4T), you can take T common out, which will give T*(T^2 - 4).
(T^2 - 4) = (T-2)(T+2)
Hence the denominator can be written as:
T*(T - 2)*(T + 2),
which makes it simple to strike the (T - 2) common between numerator and denominator.

Answer 5

Because it helps you solve the problem.
The bottom part can be factored to T(T-2)(T+2), so you can cancel the (T-2) portions, which reduces the problem to a quadratic.

Answer 6

bcoz (t-2) is the cube root of the equation t^3 - 4t
and you simplify the equation to
t^2 + 5t -2 all over (t-2)^2