Key Concepts and Formulas

• Factoring Polynomials: Identifying and extracting common factors to simplify equations.
• Substitution: Replacing a complex expression with a simpler variable to reduce the degree of the polynomial.
• Roots of Polynomials: Finding the values that satisfy the polynomial equation.
• Magnitude of Complex Numbers: For a complex number $z = a + bi$, the magnitude is $|z| = \sqrt{a^2 + b^2}$.

Step-by-Step Solution

Step 1: Factor out the common term. The given equation is: $x^7 + 3x^5 - 13x^3 - 15x = 0$ We observe that $x$ is a common factor. $x(x^6 + 3x^4 - 13x^2 - 15) = 0$ This gives us one root $x = 0$.

Step 2: Reduce the polynomial degree using substitution. The remaining polynomial equation is: $x^6 + 3x^4 - 13x^2 - 15 = 0$ We substitute $t = x^2$. $(x^2)^3 + 3(x^2)^2 - 13(x^2) - 15 = 0$ $t^3 + 3t^2 - 13t - 15 = 0$

Step 3: Find the roots of the cubic equation in $t$. The cubic equation is: $t^3 + 3t^2 - 13t - 15 = 0$ We test for rational roots. Trying $t = -1$: $(-1)^3 + 3(-1)^2 - 13(-1) - 15 = -1 + 3 + 13 - 15 = 0$ Thus, $t = -1$ is a root, and $(t+1)$ is a factor. Performing polynomial division: $(t^3 + 3t^2 - 13t - 15) \div (t+1) = t^2 + 2t - 15$ Factoring the quadratic: $t^2 + 2t - 15 = (t+5)(t-3) = 0$ The roots for $t$ are $t = -1, -5, 3$.

Step 4: Substitute back to find the roots of $x$. We have $t = x^2$.

• If $x^2 = -1$, then $x = \pm i$.
• If $x^2 = -5$, then $x = \pm i\sqrt{5}$.
• If $x^2 = 3$, then $x = \pm \sqrt{3}$. Including the root $x=0$, the roots are: $S = \{0, i, -i, i\sqrt{5}, -i\sqrt{5}, \sqrt{3}, -\sqrt{3}\}$

Step 5: Order the roots by their magnitudes.

• $|0| = 0$
• $|i| = |-i| = 1$
• $|i\sqrt{5}| = |-i\sqrt{5}| = \sqrt{5} \approx 2.236$
• $|\sqrt{3}| = |-\sqrt{3}| = \sqrt{3} \approx 1.732$ Ordering the magnitudes: $\sqrt{5}, \sqrt{5}, \sqrt{3}, \sqrt{3}, 1, 1, 0$. Assigning the roots:
• $\alpha_1, \alpha_2 = i\sqrt{5}, -i\sqrt{5}$
• $\alpha_3, \alpha_4 = \sqrt{3}, -\sqrt{3}$
• $\alpha_5, \alpha_6 = i, -i$

Step 6: Evaluate the required expression. The expression is: $\alpha_1\alpha_2 - \alpha_3\alpha_4 + \alpha_5\alpha_6$

• $\alpha_1\alpha_2 = (i\sqrt{5})(-i\sqrt{5}) = -i^2(5) = 5$
• $\alpha_3\alpha_4 = (\sqrt{3})(-\sqrt{3}) = -3$
• $\alpha_5\alpha_6 = (i)(-i) = -i^2 = 1$ So, the expression becomes: $5 - (-3) + 1 = 5 + 3 + 1 = 9$ Since the correct answer given is 7, there must be an error in the problem statement or the provided answer. Let's re-evaluate the expression assuming the answer is correct.

Suppose the correct answer is 7. Then: $5 - (-3) + \alpha_5\alpha_6 = 7$ $8 + \alpha_5\alpha_6 = 7$ $\alpha_5\alpha_6 = -1$ If $\alpha_5 = 1$ and $\alpha_6 = -1$, then the roots with magnitude 1 are 1 and -1 instead of i and -i.

If $\alpha_5, \alpha_6 = 1, -1$ then, the roots are: $S = \{0, 1, -1, i\sqrt{5}, -i\sqrt{5}, \sqrt{3}, -\sqrt{3}\}$ The magnitudes would be: $\sqrt{5}, \sqrt{5}, \sqrt{3}, \sqrt{3}, 1, 1, 0$.

Then, $\alpha_1\alpha_2 - \alpha_3\alpha_4 + \alpha_5\alpha_6 = 5 - (-3) + (1)(-1) = 5+3-1 = 7$

The only way to get the answer 7 is if $\alpha_5, \alpha_6 = 1, -1$. But in this case $x^6 + 3x^4 - 13x^2 - 15 = (x^2+5)(x^2-3)(x^2+1)$ is wrong. It should be $(x^2+5)(x^2-3)(x^2-1)$ so $x^6 + x^4 - 15x^2 + 15 = x^6 + 3x^4 - 13x^2 -15$.

Let's assume the question meant ${x^7} + 3{x^5} - 13{x^3} - 15x = x(x^2-1)(x^2+1)(x^2+5)(x^2-3)$ which is incorrect too.

Let's assume the roots are correct and the expression is different. If $\alpha_1\alpha_2 + \alpha_3\alpha_4 + \alpha_5\alpha_6 = 7$ then $5 + (-3) + 5 = 7$ So, $\alpha_1\alpha_2 + \alpha_3\alpha_4 + \alpha_5\alpha_6 = 3$

The correct expression is $\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6 = 5-(-3)+1 = 9$

Let's assume the roots are incorrect, but the question is correct. If the roots are $0, \sqrt{3}, -\sqrt{3}, 1, -1, 2, -2$ then The magnitudes would be $2, 2, \sqrt{3}, \sqrt{3}, 1, 1, 0$. Then the answer could be 4-(-3) + (-1) = 7

Common Mistakes & Tips

• Be careful with signs, especially when dealing with complex numbers and substitution.
• Remember to include all roots, including the root $x=0$ obtained from factoring.
• Double-check the magnitudes of complex numbers to ensure correct ordering.

Summary The problem involves solving a polynomial equation by factoring and substitution, finding the roots, ordering them by magnitude, and evaluating an expression. The correct evaluation of the expression with the derived roots gives 9. Since the provided correct answer is 7, there may be an error in the original question or the given answer.

The final answer is \boxed{9}.