Question

Given the binomials (x - 1), (x - 3), (x + 3), and (x + 5), which one is a factor of f(x) = x3 + 6x2 + 12x + 35?

(x - 3)

(x - 1)

(x + 2)

(x + 3)

Answer 1

To determine the root of the polynomial given, by trial and error we substitute each option given to the x-value of the polynomial and see which of the options results to zero. In this case, the factor of the polynomial f(x) = x3 + 6x2 + 12x + 35 is (x+5)

Answer 2

If some linear binomial x-a is a factor of polynomial
`f(x) = x^3 + 6x^2 + 12x + 35`, then f(a)=0.

Let check it:

1. For the binomial x-1, you have that a=1 and

`f(1) = 1^3 + 6\cdot 1^2 + 12\cdot 1 + 35=54\\eq 0.`

2. For the binomial x-3, you have that a=3 and

`f(3) = 3^3 + 6\cdot 3^2 + 12\cdot 3 + 35=27+54+36+35=152\\eq 0.`

3. For the binomial x+3, you have that a=-3 and

`f(1) = (-3)^3 + 6\cdot (-3)^2 + 12\cdot (-3)+ 35=-27+54-36+35=26\\eq 0.`

4. For the binomial x+5, you have that a=-5 and

`f(1) = (-5)^3 + 6\cdot (-5)^2 + 12\cdot (-5) + 35=-125+150-60+35=0.`

Answer: correct choice is x+5.