algebra 2 problem

Question

f(x) =x⁴+3x³-3x-9  

 

I require the real zero of f between - ? and ?

Answer 1

Let's factor it first. In this case, we can factor by grouping.

 

x⁴ + 3x³ - 3x - 9

 

= (x⁴ + 3x³) - (3x + 9)

 

= x³(x + 3) - 3(x + 3)

 

= (x³ - 3)(x + 3)

 

Setting each factor to equal to 0, the two real roots are therefore x = -3 and x = ³√3 ≈ 1.4422

 

Since your lower and upper bound are showing up as question marks in your question. You should check these two answers against the interval your teacher specified on your homework problem to see which one(s) to keep.

Answer 2

To find the zeros, we set f(x) equal to zero.

 

0 = x⁴ + 3x³ - 3x - 9

 

 

Factor f(x) by grouping.

 

0 = x³(x + 3) - 3(x + 3)

 

0 = (x³ - 3)(x + 3)

 

 

Set the factors equal to zero.

 

x³ - 3 = 0            and             x + 3 = 0

 

x³ = 3                  and            x = -3

 

x = ³√3

 

 

You have 2 real zeros:

 

x = (-3, 0)

x = (³√3, 0)