To solve for the zeros of the polynomial [tex]\( f(x) = 4x^5 + 35x^4 + 99x^3 + 33x^2 - 223x + 52 \)[/tex], follow these steps:
1. Identify the Polynomial:
The given polynomial is:
[tex]\[
f(x) = 4x^5 + 35x^4 + 99x^3 + 33x^2 - 223x + 52
\][/tex]
2. Finding the Zeros:
Zeros of a polynomial are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. The roots of this polynomial can be both real and complex.
3. Listing the Roots:
For the polynomial [tex]\( f(x) = 4x^5 + 35x^4 + 99x^3 + 33x^2 - 223x + 52 \)[/tex], the roots are:
[tex]\[
x = -4, \quad x = \frac{1}{4}, \quad x = 1, \quad x = -3 - 2i, \quad x = -3 + 2i
\][/tex]
4. Real and Complex Roots:
Among the solutions, we distinguish between real and complex roots.
- The real roots are the ones without any imaginary parts, which are:
[tex]\[
x = -4, \quad x = \frac{1}{4}, \quad x = 1
\][/tex]
- The complex roots have an imaginary part:
[tex]\[
x = -3 - 2i, \quad x = -3 + 2i
\][/tex]
5. Summary of the Roots:
- All roots of the polynomial [tex]\( f(x) = 4x^5 + 35x^4 + 99x^3 + 33x^2 - 223x + 52 \)[/tex] are:
[tex]\[
x = -4, \quad x = \frac{1}{4}, \quad x = 1, \quad x = -3 - 2i, \quad x = -3 + 2i
\][/tex]
- The real roots are:
[tex]\[
x = -4, \quad x = \frac{1}{4}, \quad x = 1
\][/tex]
In conclusion, the polynomial [tex]\( f(x) = 4x^5 + 35x^4 + 99x^3 + 33x^2 - 223x + 52 \)[/tex] has five roots in total, out of which three are real [tex]\((-4, \frac{1}{4}, 1)\)[/tex] and two are complex [tex]\((-3 - 2i, -3 + 2i)\)[/tex].
