How do you factor (2x^3 -12x^2 +10x +12)/2x -4 ?

2x³ - 12x² + 10x + 12
= 2(x³ - 6x² + 5x + 6)

Let f(x) = x³ - 6x² + 5x + 6
f(2) = (2)³ - 6(2)² + 5(2) + 6 = 0
According to Factor Theorem, (x - 2) is a factor of f(x).

2x³ - 12x² + 10x + 12
= 2(x³ - 6x² + 5x + 6)
= 2[(x³ - 2x²) + (-4x² + 8x) + (-3x + 6)]
= 2[x²(x - 2) - 4x(x - 2) - 3(x - 2)]
= 2(x - 2)(x² - 4x - 3)
= 2(x - 2)(x - 4)(x + 1)
= (2x - 4)(x - 4)(x + 1)


2x - 4
= 2(x - 2)

Hence, (2x³ - 12x² + 10x + 12) / (2x - 4)
= (2x - 4)(x - 4)(x + 1) / (2x - 4)
= (x - 4)(x + 1)

with a calculator.

= 2x³ - 12x² + 10x + 12

= 2x³ - (8x² + 4x²) + (16x - 6x) + 12

= 2x³ - 8x² - 4x² + 16x - 6x + 12

= 2x³ - 8x² - 6x - 4x² + 16x + 12

= (2x³ - 8x² - 6x) - (4x² - 16x - 12)

= x.(2x² - 8x - 6) - 2.(2x² - 8x - 6)

= (x - 2).(2x² - 8x - 6)

= 2.(x - 2).(x² - 4x - 3) ← memorize this result as (1)


= 2x - 4

= 2.(x - 2) ← memorize this result as (2)



= (2x³ - 12x² + 10x + 12) / (2x - 4) → recall (1)

= 2.(x - 2).(x² - 4x - 3) / (2x - 4) → recall (2)

= 2.(x - 2).(x² - 4x - 3) / [2.(x - 2)]

= (x - 2).(x² - 4x - 3) / (x - 2)

= x² - 4x - 3 → to go further

= x² - 4x + (4 - 4) - 3

= x² - 4x + 4 - 4 - 3

= x² - 4x + 4 - 7

= (x - 2)² - (√7)² → you recognize: a² - b² = (a + b).(a - b)

= (x - 2 + √7).(x - 2 - √7)

(2x^3 - 12x^2 + 10x + 12)/2x - 4

= (2x^3 - 12x^2 + 10x + 12 - 8x) / 4

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

= 2(x^3 - 6x^2 + x + 6) / 4

= (x^3 - 6x^2 + x + 6) / 2. That's as far as you can get, and this is provided that x is nonzero. If the whole denominator is 2x - 4:

(2x^3 - 12x^2 + 10x + 12) / (2x - 4)

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

= 2(x - 2)(x^2 - 4x - 3) / 2(x - 2)

= x^2 - 4x - 3. This is not factorable among the rational numbers, but it has irrational real roots. This is provided that x is not 2 (you'd get 0/0, which is indeterminate, and thus a hole would exist).

Take whatever answer you need according to what the denominator is in the original problem.

Hello,

Three steps are described: divide by a common number, see if a common expression can be pulled out of the complex expression in the numerator, and then divide again. Hope it helps!

(2x^3 -12x^2 +10x +12)/(2x -4)=(x^3 -6x^2 +5x +6)/(x -2)=x^2-4x-3

2(x -2)(x^2 -4x -3) / 2(x -2)