choose all true statements about the zeros of:

Question

f(x)=x⁴+7x³-12x²+40x-15

1. f(x) has 3 or 1 positive real zeros
2. 1,3, 5, 15 are the possible rational zeros
3. f(x) has 4,2 or 0 real zeros
4. f(x) has 2 or 0 negative real zeros
5. f(x) has 4 total zeros, real or complex

Answer 1

1. f(x) has 3 or 1 positive real zeros True by Descartes Rule of Signs
2. 1,3, 5, 15 are the possible rational zeros True by Rational Root Theorem
3. f(x) has 4,2 or 0 real zeros True by complex roots come in conjugate pairs
4. f(x) has 2 or 0 negative real zeros False by Descartes Rule of Signs
5. f(x) has 4 total zeros, real or complex True by the degree of the polynomial is the number of zeros

Answer 2

last one is clearly correct, 4 zeroes, real or imaginary.

actually "complex" means either real or imaginary

since they come in conjugate pairs, the 1st one looks false, it should be even numbers, not odd

1 are not roots, or zeroes, so 2nd is false, 5 and 15 just make it even further from f(x) =0

the 4,2,0 or 2,0 might be true, any 4th degree has 4 zeroes, but some may be repetitive, so it's ambiguous whether these answers are counting a repetitive zero as 1 or 1 zeroes. 2 or 0 negative roots looks correct.

4,2 or 0 real roots depends on whether all 4 are real.

the 4th degree takes 3 turns, with turning points at local max and min, found by taking the derivative and setting it = 0, if the local maximum is negative, then there's only 2 real roots. If local max >0, there are 4 real roots or zeroes.

f(0)<0

f(1)>0

so there's a real root between 0 and 1

f(-1) =<<0, very negative, but eventually the 4th degree term will dominate and it will turn positive, leaving at least on more real root