For each cubic function below, one of the zeros is given. Express each cubic function in terms of linear factors. a) f (x) = 2x³-9x²-53x-24; f (8) = 0b) f (x) = x³+x²+6x+6; f (-1) = 0

a) 2x³-9x²-53x-24 = (x-8) (2x + 1) (x + 1)

b) x³+x²+6x+6 = (x + 1) (x - i√6) (x + i√6)

Step-by-step explanation:

a)

f (x) = 2x³-9x²-53x-24

f (8) = 0

It means that 8 is the root of the function f (x)

2x³-9x²-53x-24 = (x-8) (2 x² + 7 x + 3)

Now find the factor of (2 x² + 7 x + 3)

2 x² + 7 x + 3 = 2 x² + 6 x + x + 3

= 2 x (x + 3) + 1 (x+3)

= (2x + 1) (x + 1)

So

2x³-9x²-53x-24 = (x-8) (2x + 1) (x + 1)

b)

f (x) = x³+x²+6x+6

f (-1) = 0

It means that - 1 is the root of the function f (x)

x³+x²+6x+6 = (x + 1) (x² + 6)

We know that

a² - b² = (a+b) (a-b)

i² = - 1

So

x² + 6 = (x - i√6) (x + i√6)

x³+x²+6x+6 = (x + 1) (x² + 6)

x³+x²+6x+6 = (x + 1) (x - i√6) (x + i√6)