Question

Use factor theorem to factorize 6x^3+ 17x^2+ 4x- 12 completely ​

Answer 1

Answer:

(x + 2) (2x + 3)(3x - 2)

Step-by-step explanation:

We have,

p(x) = 6x³ + 17x² + 4x - 12

Put x = 1 in p(x),

p(1) = 6(1)³ + 17(1)² + 4(1) - 12

p(1) = 6 + 17 + 4 - 12

p(1) = 15 ≠ 0

Put x = - 1 in p(x),

p(-1) = 6(-1)³ + 17(-1)² + 4(-1) - 12

p(-1) = -6 + 17 - 4 - 12

p(-1) = -22 + 17

p(-1) = -5 ≠ 0

Put x = -2 in p(x),

p(-2) = 6(-2)³ + 17(-2)² + 4(-2) - 12

p(-2) = -48 + 68 - 8 - 12

p(-2) = -68 + 68 = 0

x + 2 is a factor of p(x)

Now, 6x³ + 17x² + 4x - 12

6x³ + 12x² + 5x² + 10x - 6x + 12

6x²(x + 2) + 5x(x + 2) - 6(x + 2)

(x + 2) (6x² + 5x - 6)

(x + 2) (6x² + 9x - 4x - 6)

(x + 2) [3x(2x + 3) - 2(2x + 3)]

∴ (x + 2) (2x + 3)(3x - 2)