The polynomial is given to be:
[tex]x^4+2x^3-8x^2+16x-32[/tex]
The Rational Roots Test (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose we have some polynomial P(x) with integer coefficients and a nonzero constant term. If p represents the factors of constant term and q represents the factors of the leading coefficient, the possible rational roots will be:
[tex]\Rightarrow\frac{p}{q}[/tex]
From the polynomial given, the constant term is -32 and the leading coefficient is 1.
Therefore, the factors of the constant will be:
[tex]p=1,2,4,8,16,32[/tex]
and the factors of the leading coefficient will be:
[tex]q=1[/tex]
Hence, the possible rational roots will be:
[tex]\begin{gathered} \pm\frac{p}{q}=\pm\frac{1}{1},\pm\frac{2}{1},\pm\frac{4}{1},\pm\frac{8}{1},\pm\frac{16}{1},\pm\frac{32}{1} \\ \therefore \\ \pm\frac{p}{q}=\pm1,\pm2,\pm4,\pm8,\pm16,\pm32 \end{gathered}[/tex]
