Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the unknown variable. $$ t^{10}-2 t^{5}+1=0 $$

Step-by-Step Solution

Answer

The real solution is \( t = 1 \).

Step 1: Identify Substitution Opportunity

Notice that the expression involves a power term, specifically, it can be expressed in terms of squares. Set substitution such that \( u = t^5 \). This transforms the equation into a quadratic form: \( u^2 - 2u + 1 = 0 \).

Step 2: Simplify the Quadratic Equation

The transformed equation \( u^2 - 2u + 1 = 0 \) resembles the form \((u - 1)^2 = 0\). Therefore, the solution to the quadratic is \( u = 1 \).

Step 3: Solve the Substitution Expression

We have \( u = t^5 \) and \( u = 1 \). Substitute back to get \( t^5 = 1 \).

Step 4: Find the Solutions for the Original Variable

Solve \( t^5 = 1 \). The solutions are the fifth roots of unity. For real numbers, the solution is \( t = 1 \). Complex solutions can be found using de Moivre's theorem, giving roots as \( t = e^{2krac{ ext{pi}i}{5}} \), where \( k = 0, 1, 2, 3, 4 \).

Step 5: Conclude with the Real Solution

Since only real solutions are required, the principal real root is identified as \( t = 1 \).

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that involve sums of powers of variables with coefficients. They are an extension of linear equations and include higher degree terms. For example, a polynomial equation of degree 10 might look like \( t^{10} - 2t^5 + 1 = 0 \). Each of these terms corresponds to a power of the variable \( t \). Polynomial equations can be used to model a variety of real-world problems because they can describe complex relationships between variables.

• The degree of the polynomial is determined by the highest power of the variable.
• Solutions of polynomial equations are called roots.
• Roots can be real or complex numbers.

Understanding polynomial equations is crucial because they enable us to find values that satisfy the given mathematical conditions. In many cases, they can be transformed using different methods to find the solutions easily.

Substitution Method

The substitution method is a technique used to simplify and solve equations, particularly useful for polynomial equations. This involves replacing a complex expression with a simpler variable, making the equation easier to solve. For the given equation \( t^{10} - 2t^5 + 1 = 0 \), substituting \( u = t^5 \) reduces it to a more manageable quadratic form, \( u^2 - 2u + 1 = 0 \).The steps are:

• Identify complex terms that can be replaced or simplified.
• Introduce a substitution variable such as \( u \) to replace the identified terms.
• Solve the simpler form of the equation for the substitution variable.
• Substitute back to find the solutions for the original variable.

Substitution not only simplifies solving polynomial equations but also helps identify relationships between variables that may not be initially obvious. This method is particularly useful when dealing with high-degree polynomials and large power terms.

Roots of Unity

Roots of Unity are solutions to the equation \( t^n = 1 \), where \( n \) is a positive integer. These roots are not just simple numbers; they have geometric properties that are expressed in terms of complex numbers. For the equation \( t^5 = 1 \), the roots of unity are the fifth roots of unity.Key concepts of roots of unity include:

• There are \( n \) distinct \( n \)-th roots of unity, found using exponential notation \( t = e^{\frac{2\pi ki}{n}} \), where \( k \) ranges from 0 to \( n-1 \).
• Geometrically, these roots form a regular polygon on the complex plane, centered at the origin, on the unit circle.
• For real numbers, \( k = 0 \) gives the principal root, \( t = 1 \).
• The concept is essential for understanding cyclic phenomena and periodic functions in mathematics and engineering.

Using roots of unity helps find solutions to polynomial equations that would be otherwise difficult to solve using real numbers alone. This provides deep insight into complex numbers and their utilitarian role in advanced mathematics.