Question

Consider the following.f(x) = -2x4 + 13x3 – 21x2 + 2x + 8(a) List the possible rational zeros of f. (Enter your answers as a comma-separated list.

Answer 1

$\operatorname{\pm}1,\operatorname{\pm}\frac{1}{2},\operatorname{\pm}2,\operatorname{\pm}4,\operatorname{\pm}8$

1) We need to use the Rational Roots Theorem:

$f\mleft(x\mright)=-2x^4+13x^3-21x^2+2x+8$

So, let's write out a ratio with the possible rational roots, considering the constant term and the leading coefficient:

$\frac{constant\:term\:divisors}{leading\:coefficient\:divisors}=\frac{\pm1,2,4,8}{\pm1,2}$

a) So the list of all possible zeros is:

$\pm1,\pm\frac{1}{2},\pm2,\pm4,\pm8$

Note that "possible' does not mean the list of all real rational roots. We'd need to plug into the function and check