F4 Maths Factor Theorem 3

1.Let f(x) =x^4+x^3-8x^2+2x+4

f(1) = 1+1-8+2+4 = 0 , so x-1 is a factor
f(2) = 16 + 8 - 32 + 4 + 4 = 0 , so x-2 is a factor.

x^4+x^3-8x^2+2x+4=0

(x - 1)(x - 2)(x^2 + 4x + 2) = 0

x = 1 ,or x = 2 , for x - 1 = 0 or x - 2 = 0

when x^2 + 4x + 2 = 0

x = [- 4 +/- sqrt(16 - 4*2) ] / 2

or x = - 2 +/- sqrt 2

2. x(x+1)(x+2)=3．4．5

Let x+1 be y:

(y-1) y (y+1) = (y^3 - y) = 60

y^3 - y - 60 = 0 = f(y)

f(4) = 4^3 - 4 - 60 = 0 , so (y-4) is a factor

(y - 4)(y^2 + 4y + 15) = 0

y = 4 when y-4 = 0,
△= 4^2 - 4*15 < 0 for y^2 + 4y + 15 = 0

y = 4 as
x = y-1 = 4-1 = 3 (triple roots)

3. 3x^3-2x^2-12x+8=0

Let f(x) = 3x^3-2x^2-12x+8

f(2) = 3*2^3 - 2(3)^2 - 12*2 + 8 = 0 , so x - 2 is a factor.

(x - 2)(3x^2 + 4x - 4) = 0

(x - 2)(3x - 2)(x + 2) = 0

x = 2 or x = 2/3 or x = - 2