To find all the real zeros of the polynomial [tex]\( P(x) = x^4 - 9x^3 + 10x^2 + 27x + 7 \)[/tex], we need to solve the equation [tex]\( P(x) = 0 \)[/tex]. Here’s a detailed, step-by-step solution:
1. Identify the polynomial:
[tex]\[ P(x) = x^4 - 9x^3 + 10x^2 + 27x + 7 \][/tex]
2. Solve [tex]\( P(x) = 0 \)[/tex] for [tex]\( x \)[/tex]:
We are looking for the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( x^4 - 9x^3 + 10x^2 + 27x + 7 = 0 \)[/tex].
3. Find rational roots:
We check if there are any obvious rational roots using techniques like the Rational Root Theorem. For brevity, let's proceed with the result:
4. Factorize or use other methods to find roots:
By solving the polynomial equation, we obtain the following roots for [tex]\( x \)[/tex]:
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 7 \)[/tex]
- [tex]\( x = \frac{3}{2} - \frac{\sqrt{13}}{2} \)[/tex]
- [tex]\( x = \frac{3}{2} + \frac{\sqrt{13}}{2} \)[/tex]
5. Verification:
To confirm these roots, you can substitute each back into the polynomial [tex]\( P(x) \)[/tex] to ensure [tex]\( P(x) = 0 \)[/tex].
Hence, the real zeros of the polynomial [tex]\( P(x) = x^4 - 9x^3 + 10x^2 + 27x + 7 \)[/tex] are:
[tex]\[ x = -1, 7, \frac{3}{2} - \frac{\sqrt{13}}{\2}, \frac{3}{2} + \frac{\sqrt{13}}{\2} \][/tex]
These are all the real zeros of the given polynomial.
