IDNLearn.com makes it easy to get reliable answers from knowledgeable individuals. Discover prompt and accurate answers from our experts, ensuring you get the information you need quickly.
Sagot :
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.
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.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.