Answer
The zeros of the polynomial function using the rational zero theorem is
[tex]\frac{\pm p}{q}=\pm1,\pm\frac{1}{2},\pm\frac{1}{4},\pm2,\pm4[/tex]
Explanation
The given polynomial function is
[tex]f(x)=4x^4+8x^3+21x^2+17x+4[/tex]
What to find:
To find the zeros of the polynomial function the rational zero theorem.
Step-by-step solution:
The rational zero theorem: If a polynomial function, written in descending order of the exponents, has integer coefficients, then any rational zero must be of the form ± p/ q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Considering the given polynomial function
[tex]f(x)=4x^4+8x^3+21x^2+17x+4[/tex]
The constant term, p = 4
The leading coefficient, q = 4
The factors of the constant p and the leading coefficient q are:
[tex]\begin{gathered} p=\pm1,\pm2,\pm4 \\ \\ q=\operatorname{\pm}1,\operatorname{\pm}2,\operatorname{\pm}4 \end{gathered}[/tex]
Hence, the zeros of the polynomial function using the rational zero theorem will be
[tex]\begin{gathered} \frac{\pm p}{q}=\frac{\pm1,\pm2,\pm4}{\pm1,\pm2,\pm4} \\ \\ \frac{\operatorname{\pm}p}{q}=\operatorname{\pm}1,\operatorname{\pm}\frac{1}{2},\operatorname{\pm}\frac{1}{4},\operatorname{\pm}2,\operatorname{\pm}4 \end{gathered}[/tex]
