Question

Let \(\omega=\frac{-1+i \sqrt{3}}{2}\), and \(\omega, \omega, \alpha_{1}, \alpha_{2}, \ldots, \alpha_{8}\) be roots of equation \(x^{10}+a x+b=0\), where \(\alpha_{1} \neq \alpha_{i}, i \neq j\) value of \(\left(\omega-\alpha_{1}\right)\left(\omega-\alpha_{2}\right) \ldots .\left(\omega-\alpha_{8}\right)\) \(=\) (1) \(90 \omega^{2}\) (2) \({ }^{10} \mathrm{C}_{2} \omega^{2}\) (3) \(10 \omega+a\) (4) 0

Short answer

0.

Step-by-step solution

Identify roots

Recognize that \(x^{10} + ax + b = 0\) has roots \( \omega, \omega, \alpha_1, \alpha_2, \ldots, \alpha_8 \) where \(\omega = \frac{-1 + i\sqrt{3}}{2}.\)

Use Vieta's Formulas

Using Vieta’s formulas, the sum of all roots taken one at a time for the polynomial equation \(x^{10} + ax + b = 0\) is zero. Hence, \[2 \omega + ( \alpha_1 + \alpha_2 + \ldots + \alpha_8 ) = 0.\]

Simplify for other roots

Since \[2 \omega + ( \alpha_1 + \alpha_2 + \ldots + \alpha_8 ) = 0,\] we can say \[ \alpha_1 + \alpha_2 + \ldots + \alpha_8 = -2 \omega.\]

Simplify expression

To find the value of \( (\omega - \alpha_1)(\omega - \alpha_2) \ldots (\omega - \alpha_8),\) substitute \[2\omega = 1,\] which simplifies the roots further. Noticing \(2\omega eq 1,\) simplifies the roots by canceling terms involving \omega.

Calculate the product of differences

Simplify the Catalan series \(y\)

Evaluate result

Afterward, the overall results cancel the final solution to be zero.

Key concepts

Complex Numbers

Complex numbers extend the idea of one-dimensional real numbers to the two-dimensional complex plane by using imaginary units. A complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

In the given exercise, we encounter the complex number \(\omega = \frac{-1 + i\sqrt{3}}{2}\). This form involves both a real part and an imaginary part. Such numbers are crucial in solving polynomial equations that do not have solely real roots.

Understanding the algebraic operations on complex numbers, such as addition, subtraction, multiplication, and division, is essential. Operations follow specific rules, especially when involving the imaginary unit \(i\). For example, \( (a + bi) + (c + di) = (a + c) + (b + d)i \).

Conjugation is another important concept. The conjugate of a complex number \(a + bi\) is \(a - bi\), and this is significant in simplifying expressions and solving equations.

Vieta's Formulas

Vieta's formulas are a set of equations relating the coefficients of a polynomial to sums and products of its roots. They are named after François Viète, a French mathematician.

For a polynomial \(P(x) = x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\), Vieta's formulas link the coefficients \(a_i\) to the roots of the polynomial. For example, for a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(x_1\) and \(x_2\) is given by \(-b/a\), and the product is \(c/a\). Similar relations hold for higher-degree polynomials.

In the problem \(x^{10} + ax + b = 0\), the sum of the roots taken one at a time is related to the coefficient of \(x^9\), which is zero because there is no \(x^9\) term. Therefore, the sum \(2\omega + (\alpha_1 + \alpha_2 + \ldots + \alpha_8) = 0\). This allows us to determine that \(\alpha_1 + \alpha_2 + \ldots + \alpha_8 = -2\omega\). Using these sums and products efficiently can simplify the solution of polynomial equations.

Polynomial Equations

Polynomial equations are algebraic expressions that set a polynomial equal to zero. They are fundamental in algebra and frequently arise in various mathematical problems. A polynomial of degree \(n\) can be written as \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_i\) are coefficients.

In our exercise, we have a 10th-degree polynomial equation: \(x^{10} + ax + b = 0\). Solving this equation involves finding the values of \(x\) (called roots) that satisfy the equation.

There are different methods to solve polynomial equations, especially higher-degree ones, such as factoring, using the Rational Root Theorem, synthetic division, and numerical methods. Each root can be real or complex, and the Fundamental Theorem of Algebra guarantees that a polynomial of degree \(n\) has exactly \(n\) roots, counting multiplicities.

In the context of this problem, after identifying the roots \(\omega\) and additional distinct roots \(\alpha_1, \alpha_2, \ldots, \alpha_8\), we use properties of polynomial roots and Vieta's formulas to find that the product \((\omega - \alpha_1)(\omega - \alpha_2) \ldots (\omega - \alpha_8)\) simplifies to zero, which aligns with option (4) in the given choices.