How do find the roots for these functions?

Question

A) f(x)=x^3+6x^2-7x

 

B) g(x)= 2x^3-250

 

 

 

Answer 1

Hey there Bat

 

If we can factor these then we can set each factor equal to zero and have the roots

 

let's look at the first one

 

first factor out an x

 

x(x^2+6x-7)

 

then factor the part in parenthesis

 

x(x+7)(x-1)

 

set each factor equal to zero and solve for x

 

x=0   and    x+7=0   and    x-1=0

x=0                x=-7             x=1

 

 

so the roots are 0, -7 and 1

 

 

Now for the 2nd one

 

first factor out the 2

2(x^3-125)

 

then the part in parenthesis

 

that is a difference of cubes

so factor it using the rule for difference of cubes

 

2(x-5)(x^2+5x+25)

 

 

the part in the last parenthesis will not factor so we can use quadratic formula

 

-5 +-SQRT(5^2-4(1)(25)) all over 2

 

that makes -5 +- sqrt (-75) all over 2

 

that makes -5 +- 5(sqrt 3) i all over 2

 

simplifies to -5/2 +- (5sqrt3/2)i

 

so there are 2 roots and one factor was (x-5) so that gives us one more root of 5

 

and that is all 3 of them.

 

Answer 2

For part b

 

2( X^3 - 125) = 0

 

X = 5

 

Part a

 

 x (x^2+6x-7)=0

 

 x (x-7)(x+1)=0

  

 x=0, x=7 and x =-1