if f(x) has a leading term of :

Question

3x⁴ and zeros at x=2, x= -2, and x= -5, choose all possibly true statements:

1. The sign of f(x) changes twice
2. x-5 is a factor of f(x)
3. x² -4 is a factor of f(x)
4. the constant term is 4
5. 3- 2√10 is a zero of f(x)

Answer 1

From the information given, we can get the following:

(x-2) (x+2) (x+5) (3x + y)

The variable 'y' in the above factored polynomial represents the unknown piece of information. From just the above, we are able to eliminate the following options:

1. x-5 is a factor of f(x) -> the factor is x+5
2. the constant term is 4 -> not enough information to make this determination
3. 3- 2√10 is a zero of f(x) -> not enough information to make this determination

Now we are left with the remaining options which are possibly true statements:

1. The sign of f(x) changes twice -> when graphing the above polynomial with arbitrary values for 'y' we notice that the sign of f(x) changes exactly 4 times due to the 4 distinct zeros.
2. x² -4 is a factor of f(x) -> when we foil the first two factors of the polynomial (x-2)(x+2) we get x^2 - 4

Answer 2

I have the impression that we are to assume f(x) to be a polynomial.

Moreover, I will assume that all the coefficients of the polynomial are real, as opposed to complex.

But I am not assuming whether the coefficients are rational or not.

So we can write

f(x) = 3(x-2)(x+2)(x+5)(x-r),

for some r, not given.

We only know that r is real, not complex, because of the assumption

that all the coefficients of f(x) are real.

The statement of the problem has two possible interpretations:

Interpretation I: The three given roots are all the roots of f(x).

In this case r must be one of 2, -2, or -5.

Interpretation II: The fourth root, r, is none of 2, -2, -5.

1) The sign of f(x) changes twice under interpretation I and four times

under interpretation II.

This is clear by drawing the possible graphs.

2) Under interpretation I, x-5 is not a factor of f(x), because x=5 is not a root.

Under interpretation II, x-5 could be a factor, if r = 5,

but there is nothing that implies it.

3) x²-4 is a factor of f(x) under both interpretations.

This is because (x-2)(x+2) are factors.

4) Under interpretation I the constant term must be 120, -120, or -300.

This comes from considering the three possible values of r.

Inder Interpretation II the constant term can be anything, including 4.

5) Under interpretation I, 3- 2√10 cannot be a root of f(x).

Under interpretation II, any real number can be the remaining root,

including 3- 2√10.

Answer 3

1. The sign of f(x) changes twice [since you don't know the 4th zero, you can't say how many sign changes there are]
2. x-5 is a factor of f(x) [since "5" is not listed as a zero, you can't say that "x - 5" is a factor]
3. x² -4 is a factor of f(x) [since you are given that x = 2 and x = -2 are zeros, you CAN say that (x -2) and (x + 2) are factors and since they multiply together to get "x² - 4" you can say this statement is true.]
4. the constant term is 4 [there is no indication this is true, although it is possible]
5. 3- 2√10 is a zero of f(x) [this is definitely not true because if "3 - 2√10" was a zero then "3 + 2√10" would also be a zero and you are already given 3 out of the 4 possible zeros]