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A polynomial [tex]$f(x)$[/tex] and two of its zeros are given.

[tex]\[ f(x) = 4x^5 + 35x^4 + 99x^3 + 33x^2 - 223x + 52 \][/tex]

(Note: The problem states that two zeros are given, but they are not provided in the text. Please include the zeros for a complete problem statement.)


Sagot :

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].
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