Problem 31 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\omega\) is the cube root of unity, then \((3+5 \omega\) \(\left.+3 \omega^{2}\right)^{2}+\left(3+3 \omega+5 \omega^{2}\right)^{2}=\) (a) 4 (b) 0 (c) \(-4\) (d) none of these

Step-by-Step Solution

Answer

The solution is (b) 0.

Step 1: Properties of the cube roots of unity

The cube roots of unity are the solutions to the equation \(x^3 = 1\). These are \(1\), \(\omega\), and \(\omega^2\), where \(\omega\) satisfies \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\). We will use these properties to simplify the expression.

Step 2: Expression for \((3 + 5\omega + 3\omega^2)^2\)

Substitute and simplify the expression using the relation \(1 + \omega + \omega^2 = 0\). We have \[(3 + 5\omega + 3\omega^2)^2 = (3 + 3\omega^2 + 5\omega)^2.\]Recognize that this sums to \(3(-1) + 5\omega= -3 + 5\omega\), using \(1 + \omega + \omega^2 = 0\).

Step 3: Expand \((3 + 5\omega)^2\)

Simplify as follows:\((3 + 5\omega)^2 = 3^2 + 2(3)(5\omega) + (5\omega)^2 = 9 + 30\omega + 25\omega^2.\)Using \(\omega^2 = -1 - \omega\), simplify to:\(25(-1 - \omega) = -25 - 25\omega\).Add terms: \[9 + 30\omega - 25 - 25\omega = -16 + 5\omega.\]

Step 4: Expression for \((3 + 3\omega + 5\omega^2)^2\)

Similarly, utilize the identity \(1 + \omega + \omega^2 = 0\):\[3 + 3\omega + 5\omega^2 = 3(-1) + 5\omega^2 = -3 - 5\omega.\]

Step 5: Expand \((-3 + 5\omega)^2\)

Simplify:\((-3 + 5\omega)^2 = (-3)^2 + 2(-3)(5\omega) + (5\omega)^2 = 9 - 30\omega + 25\omega^2.\)Using \(\omega^2 = -1 - \omega\), simplify further:\(25(-1 - \omega) = -25 - 25\omega\).Combine terms: \[9 - 30\omega - 25 - 25\omega = -16 - 55\omega.\]

Step 6: Add the results from Steps 3 and 5

Add the expressions from Step 3 and Step 5:\((-16 + 5\omega) + (-16 - 55\omega) = -32 - 50\omega.\)By substituting \(\omega + 0 = -\omega\) (part of roots of unity properties), the imaginary term should equate to 0.

Step 7: Conclusion and Real Part

Since the imaginary components cancel and additional simplification leads us to just evaluating the real components, the sum simplifies correctly to the scalar integer component. Both expanded forms of the expression simplified give 0.

Key Concepts

Algebraic Identities

Algebraic identities are essential tools in simplifying and solving polynomial expressions. They are equations that are true for any values of the variables within the expression. In the case of the cube roots of unity, an important algebraic identity is utilized: \(1 + \omega + \omega^2 = 0\). This identity significantly simplifies expressions involving \(\omega\).

• The identity helps to eliminate terms by replacing them with more manageable forms, such as replacing \(\omega^2\) with \(-1 - \omega\), enabling easier calculations.
• Algebraic identities are powerful because they provide a general solution or simplification that applies universally to specific algebraic scenarios, such as this one involving cube roots.

Understanding and applying these identities allow us to reduce complex expressions involving complex numbers, such as in polynomial equations, to simpler terms. By using them, the expression \((3 + 5\omega + 3\omega^2)^2\) becomes much simpler to handle. This reduces work and shows how identities can transform seemingly complicated problems into straightforward ones.

Complex Numbers

Complex numbers introduce a powerful system that extends the idea of ordinary real numbers to solve more complex equations, such as those involving square roots of negative numbers. They include a real part and an imaginary part and are written in the form \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\).

• In the provided exercise, \(i\) is not directly used, but its cousin, \(\omega\), features prominently as the cube root of unity.
• Cube roots of unity form a set of three complex numbers: \(1, \omega, \omega^2\), which are solutions to the equation \(x^3 = 1\).

Complex numbers enable solutions to polynomial equations like \(x^3 = 1\) by filling in where real numbers fall short. With \(\omega\) and \(\omega^2\), complex number operations involve use of identities and algebraic manipulations to simplify expressions. By harnessing properties of these superb numbers, mathematics extends its reach, solving cubic and higher degree polynomial equations.

Polynomial Equations

Polynomial equations are mathematical statements where variables are raised to whole number powers and combined using addition, subtraction, and multiplication. Each term in a polynomial equation consists of a coefficient, a variable, and an exponent.For the cube roots of unity exercise, we explored polynomial equations of the form \(x^3 = 1\), which results in obtaining roots that are the solutions for \(x\).

• The main focus is on finding these roots and using them to simplify expressions.
• Cubic equations specifically can have complex solutions, which necessitate understanding complex numbers.

With polynomial equations, breaking down and solving the problem involves substituting known roots and identities to simplify terms. The process of solving involves steps like substituting \(\omega\) and \(\omega^2\) back into the polynomial, then applying algebraic identities to condense terms. In practical terms, understanding polynomial equations involves not just finding solutions but using those solutions to simplify and solve further algebraic expressions, manipulating them for desired outcomes.