Since this polynomial has 4 terms, factoring by grouping should be the first thing we try here.
So, we have:
[tex] 8x^2(3x-8)-7(3x-8)=0 \implies\\
(8x^2-7)(3x-8)=0 [/tex]
So, we can use ZPP to find out roots:
[tex] 3x-8 =0 \implies\\ x=\frac{8}{3}\\
8x^2-7=0 \implies \\
8x^2=7 \implies \\
x^2=\frac{7}{8} \implies \\
x=\pm\frac{\sqrt{7}}{\sqrt{{8}}}\\
\text{Rationalizing denominator:}\\
x=\pm\frac{\sqrt{56}}{8} [/tex]
So our three roots are:
[tex] x \in \{\frac{8}{3},\frac{\sqrt{56}}{8},-\frac{\sqrt{56}}{8}\} [/tex]
