Answer :
- The polynomial has 5 complex roots.
- The possible number of real roots are 5, 3, or 1, and imaginary roots are 0, 2, or 4.
- Possible rational roots are $\pm 1, \pm 5, \pm 7, \pm 35$.
- The roots are $5, i, -i, i\sqrt{7}, -i\sqrt{7}$.
$\boxed{{5, i, -i, i\sqrt{7}, -i\sqrt{7}}}$
### Explanation
1. Understanding the Problem
We are given the polynomial equation $x^5-5 x^4+8 x^3-40 x^2+7 x-35=0$. Our goal is to determine the number of complex roots, the possible number of real and imaginary roots, the possible rational roots, and finally, to find all the roots of the equation.
2. Finding the Number of Complex Roots
First, let's find the number of complex roots. According to the Fundamental Theorem of Algebra, a polynomial of degree $n$ has exactly $n$ complex roots, counting multiplicities. Since our polynomial has degree 5, it has 5 complex roots.
3. Determining Possible Real and Imaginary Roots
Next, we determine the possible number of real and imaginary roots. Complex roots occur in conjugate pairs (if the coefficients of the polynomial are real). This means that if $a + bi$ is a root, then $a - bi$ is also a root. Therefore, the number of imaginary roots must be even. Since the total number of roots is 5, the number of real roots can be 5, 3, or 1, and the corresponding number of imaginary roots will be 0, 2, or 4.
4. Listing Possible Rational Roots
Now, let's find the possible rational roots. We use the Rational Root Theorem, which states that any rational root of the polynomial must be of the form $\pm \frac{p}{q}$, where $p$ is a factor of the constant term (-35) and $q$ is a factor of the leading coefficient (1). The factors of 35 are 1, 5, 7, and 35. Therefore, the possible rational roots are $\pm 1, \pm 5, \pm 7, \pm 35$.
5. Testing Possible Rational Roots
We can test these possible rational roots by substituting them into the polynomial or using synthetic division. Let's try $x = 5$:
$5^5 - 5(5^4) + 8(5^3) - 40(5^2) + 7(5) - 35 = 3125 - 3125 + 1000 - 1000 + 35 - 35 = 0$. So, $x = 5$ is a root.
6. Dividing the Polynomial
Now we divide the polynomial by $(x - 5)$ to find the remaining polynomial. Performing polynomial division, we get:
$\frac{x^5-5 x^4+8 x^3-40 x^2+7 x-35}{x-5} = x^4 + 8x^2 + 7$.
7. Solving the Quartic Equation
We now need to solve the quartic equation $x^4 + 8x^2 + 7 = 0$. Let $y = x^2$. Then the equation becomes $y^2 + 8y + 7 = 0$. This is a quadratic equation in $y$. We can factor it as $(y + 1)(y + 7) = 0$. So, $y = -1$ or $y = -7$.
8. Finding the Roots
If $y = -1$, then $x^2 = -1$, so $x = \pm i$. If $y = -7$, then $x^2 = -7$, so $x = \pm i\sqrt{7}$. Therefore, the roots are $5, i, -i, i\sqrt{7}, -i\sqrt{7}$.
9. Final Answer
In summary:
- Number of complex roots: 5
- Possible number of real roots: 5, 3, or 1
- Possible number of imaginary roots: 0, 2, or 4
- Possible rational roots: $\pm 1, \pm 5, \pm 7, \pm 35$
- Roots: $5, i, -i, i\sqrt{7}, -i\sqrt{7}$
### Examples
Understanding polynomial roots is crucial in many areas of engineering and physics. For example, in control systems, the roots of the characteristic equation determine the stability of a system. In circuit analysis, roots can represent resonant frequencies. Finding the roots allows engineers to design stable and efficient systems.
- The possible number of real roots are 5, 3, or 1, and imaginary roots are 0, 2, or 4.
- Possible rational roots are $\pm 1, \pm 5, \pm 7, \pm 35$.
- The roots are $5, i, -i, i\sqrt{7}, -i\sqrt{7}$.
$\boxed{{5, i, -i, i\sqrt{7}, -i\sqrt{7}}}$
### Explanation
1. Understanding the Problem
We are given the polynomial equation $x^5-5 x^4+8 x^3-40 x^2+7 x-35=0$. Our goal is to determine the number of complex roots, the possible number of real and imaginary roots, the possible rational roots, and finally, to find all the roots of the equation.
2. Finding the Number of Complex Roots
First, let's find the number of complex roots. According to the Fundamental Theorem of Algebra, a polynomial of degree $n$ has exactly $n$ complex roots, counting multiplicities. Since our polynomial has degree 5, it has 5 complex roots.
3. Determining Possible Real and Imaginary Roots
Next, we determine the possible number of real and imaginary roots. Complex roots occur in conjugate pairs (if the coefficients of the polynomial are real). This means that if $a + bi$ is a root, then $a - bi$ is also a root. Therefore, the number of imaginary roots must be even. Since the total number of roots is 5, the number of real roots can be 5, 3, or 1, and the corresponding number of imaginary roots will be 0, 2, or 4.
4. Listing Possible Rational Roots
Now, let's find the possible rational roots. We use the Rational Root Theorem, which states that any rational root of the polynomial must be of the form $\pm \frac{p}{q}$, where $p$ is a factor of the constant term (-35) and $q$ is a factor of the leading coefficient (1). The factors of 35 are 1, 5, 7, and 35. Therefore, the possible rational roots are $\pm 1, \pm 5, \pm 7, \pm 35$.
5. Testing Possible Rational Roots
We can test these possible rational roots by substituting them into the polynomial or using synthetic division. Let's try $x = 5$:
$5^5 - 5(5^4) + 8(5^3) - 40(5^2) + 7(5) - 35 = 3125 - 3125 + 1000 - 1000 + 35 - 35 = 0$. So, $x = 5$ is a root.
6. Dividing the Polynomial
Now we divide the polynomial by $(x - 5)$ to find the remaining polynomial. Performing polynomial division, we get:
$\frac{x^5-5 x^4+8 x^3-40 x^2+7 x-35}{x-5} = x^4 + 8x^2 + 7$.
7. Solving the Quartic Equation
We now need to solve the quartic equation $x^4 + 8x^2 + 7 = 0$. Let $y = x^2$. Then the equation becomes $y^2 + 8y + 7 = 0$. This is a quadratic equation in $y$. We can factor it as $(y + 1)(y + 7) = 0$. So, $y = -1$ or $y = -7$.
8. Finding the Roots
If $y = -1$, then $x^2 = -1$, so $x = \pm i$. If $y = -7$, then $x^2 = -7$, so $x = \pm i\sqrt{7}$. Therefore, the roots are $5, i, -i, i\sqrt{7}, -i\sqrt{7}$.
9. Final Answer
In summary:
- Number of complex roots: 5
- Possible number of real roots: 5, 3, or 1
- Possible number of imaginary roots: 0, 2, or 4
- Possible rational roots: $\pm 1, \pm 5, \pm 7, \pm 35$
- Roots: $5, i, -i, i\sqrt{7}, -i\sqrt{7}$
### Examples
Understanding polynomial roots is crucial in many areas of engineering and physics. For example, in control systems, the roots of the characteristic equation determine the stability of a system. In circuit analysis, roots can represent resonant frequencies. Finding the roots allows engineers to design stable and efficient systems.
