Question

If cube root of unity are \(1, w, w^{2}\) then the roots of the equation \((\mathrm{x}-1)^{3}+8=0\) are (a) \(-1,-1,-1\) (b) \(-1,-1+2 \mathrm{w},-1-2 \mathrm{w}^{2}\) (c) \(-1,1+2 \mathrm{w}, 1+2 \mathrm{w}^{2}\) (d) \(-1,1-2 \mathrm{w},+1-2 \mathrm{w}^{2}\)

Short answer

The short answer to the question is: (d) \(-1, 1 - 2w, 1 - 2w^2\).

Step-by-step solution

Rewrite the given equation

Rewrite the given equation \((x-1)^3 + 8 = 0\) as \[(x-1)^3 = -8\]

Express -8 as -2 times cube root of unity

Knowing that \(w^3 = 1\), we can express -8 as a product of -2 and cube root of unity: \[-8 = -2 \cdot (1 + w + w^2)\]

Make x the subject

Now, we can write x in terms of the cube roots of unity: \[(x-1)^3 = -2(1 + w + w^2) \Rightarrow x = 1 - \sqrt[3]{2(1 + w + w^2)}\]

Find the three roots

Substitute the cube roots of unity \(1, w, w^2\) into the expression for x: 1. When the cube root of unity is 1: \[x_1 = 1 - \sqrt[3]{2(1 + 1 + 1)} = 1 - 2 = -1\] 2. When the cube root of unity is w: \[x_2 = 1 - \sqrt[3]{2(1 + w + w^2)} = 1 - 2w\] 3. When the cube root of unity is \(w^2\): \[x_3 = 1 - \sqrt[3]{2(1 + w^2 + w^4)} = 1 - 2w^2\]

Compare the roots with the given options

Comparing the three roots we found, \(x_1 = -1\), \(x_2 = 1 - 2w\), and \(x_3 = 1 - 2w^2\), with the given options, we can see that the correct answer is: (d) \(-1, 1 - 2w, 1 - 2w^2\)

Key concepts

Cube Roots

Cube roots are a fundamental concept when we deal with polynomial equations of degree three. A cube root of a number is a value that, when multiplied by itself three times, gives that original number. For example, the cube root of 8 is 2 because \(2^3 = 8\).
In the realm of complex numbers, cube roots play an intriguing role. Not only do they help us solve cubic equations, but they also reveal properties that real numbers do not exhibit. We'll see that one of these properties is their connection with roots of unity, which can take us into complex numbers.
Cube roots of unity, in particular, are special because they represent the three solutions of the equation \(x^3 = 1\). These roots are 1, \(w\), and \(w^2\), where \(w\) is a primitive cube root of unity. This means that when you cube each of these roots, the result is 1.
Diving into the world of cube roots and how they interrelate with complex numbers unfolds a deeper understanding of algebraic structures.

Complex Numbers

Complex numbers extend our familiar number system by explaining phenomena that cannot be understood by real numbers alone. A complex number is generally expressed as \(a + bi\), where *a* is the real part and *b* is the imaginary part of the complex number, while \(i\) is the imaginary unit, satisfying \(i^2 = -1\).
The value \(w\) from our earlier discussion is a complex number, specifically, a root of unity. \(w\) and \(w^2\) are existing within the complex plane, providing essential solutions to polynomial equations.
Understanding these numbers can be essential for grasping higher-level math concepts and even in everyday applications like engineering and physics. They represent vectors in the complex plane and provide more flexibility than real numbers alone. Complex numbers make polynomial roots, such as in this exercise, an engaging problem to solve with unique solutions beyond the simple and real.

Algebraic Equations

Algebraic equations form the core of algebra, defining relationships between different quantities. These equations can range from simple linear equations, such as \(ax + b = 0\), to complex polynomial equations, like those involving powers and roots.
Cubic equations specifically consist of the highest term \(x^3\) and are a type of polynomial equation. They can be solved either graphically, numerically, or algebraically by factoring or using the formula for roots.
In this particular problem, we dealt with a modified cubic equation, \((x-1)^3 + 8 = 0\). Through manipulation and understanding cube roots of unity, we found the complex solutions for the variable.

• Set the equation to zero and solve for \(x\).
• Use identities involving roots of unity.
• Express the solutions in their simplest form.

The manipulation of these equations builds a foundation for even more complex mathematical ideas.

Polynomial Roots

Polynomial roots represent the values of \(x\) for which the polynomial equation equals zero. These roots are crucial in determining the behavior and characteristics of polynomial functions.
Each polynomial equation has a number of roots equal to its degree, thus a cubic equation has three roots which can be real, complex, or a mixture of both.
In solving \((x-1)^3 + 8 = 0\), we are essentially executing a search for such roots, using division and identities. The cube root of unity played a powerful role, helping us land on the solutions through the systematic approach.

• Real roots provide intersection points with the \(x\)-axis.
• Complex roots make the polynomial "turn" in the complex plane.

Understanding polynomial roots is not just about solving equations, but also about appreciating the dynamic nature of algebra and its graphical representations.