Question

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

Answer 1

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

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

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)