Question

Find the zeros of polynomial function and solve polynomials equations. f(x)=24x^3-64x^2-21x+56

Answer 1

Since this polynomial has 4 terms, factoring by grouping should be the first thing we try here.

So, we have:

$8x^2(3x-8)-7(3x-8)=0 \implies\\ (8x^2-7)(3x-8)=0$

So, we can use ZPP to find out roots:

$3x-8 =0 \implies\\ x=\frac{8}{3}\\ 8x^2-7=0 \implies \\ 8x^2=7 \implies \\ x^2=\frac{7}{8} \implies \\ x=\pm\frac{\sqrt{7}}{\sqrt{{8}}}\\ \text{Rationalizing denominator:}\\ x=\pm\frac{\sqrt{56}}{8}$

So our three roots are:

$x \in \{\frac{8}{3},\frac{\sqrt{56}}{8},-\frac{\sqrt{56}}{8}\}$