Problem 14 - Algebra 2

Question

Solve each equation. State the number and type of roots. \(x^{4}-81=0\)

Step-by-Step Solution

Answer

The equation has two real roots (3, -3) and two imaginary roots (3i, -3i).

Step 1: Rearrange the Equation

The equation given is \(x^4 - 81 = 0\). Rearrange it to isolate \(x^4\) on one side: \(x^4 = 81\).

Step 2: Apply the Square Root

Take the square root of both sides to simplify \(x^4 = 81\). This gives \((x^2)^2 = 81\). By taking the square root of both sides, we get \(x^2 = 9\) and \(x^2 = -9\).

Step 3: Solve for \(x^2 = 9\)

For \(x^2 = 9\), take the square root of both sides, leading to \(x = 3\) and \(x = -3\). These are the real roots.

Step 4: Solve for \(x^2 = -9\)

For \(x^2 = -9\), recognize that the square root of a negative number is imaginary. The solutions are \(x = 3i\) and \(x = -3i\). These are the imaginary roots.

Step 5: Determine the Number and Type of Roots

The equation has two real roots (3 and -3) and two imaginary roots (3i and -3i). Altogether, the equation has four roots.

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients. These variables are raised to non-negative integer powers. A polynomial equation like \(x^4 - 81 = 0\) involves terms like \(x^4\) and constants like -81.

• The degree of a polynomial is determined by the highest power of the variable in the equation. Here, \(x^4\) indicates a fourth-degree polynomial, as the highest exponent is 4.
• Polynomials can have different types of solutions depending on the degree. For a fourth-degree polynomial, there are generally four solutions — these could be real or complex.
• When solving polynomial equations, especially higher degree ones, factorization or transformations such as applying roots are often used to simplify and find solutions.

Understanding polynomials helps in recognizing the behavior of various equations and predicting the nature of their roots.

Complex Numbers

Complex numbers introduce a way to solve equations that do not have solutions within the set of real numbers. They are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).

• Complex numbers extend the real number system. Real numbers are actually a subset of complex numbers where the imaginary part is zero.
• In solving equations, complex numbers allow us to handle roots of negative numbers, which aren't possible within the real number system alone.
• They are crucial in advanced mathematical concepts and in the description of phenomena in engineering and physics, where real and imaginary components are used to model real-life situations.

Complex numbers expand our ability to solve equations and model complex systems, beyond what real numbers alone can achieve.

Imaginary Roots

Imaginary roots arise when we solve equations that require taking the square root of a negative number. In the context of the equation \(x^2 = -9\), the roots are \(x = 3i\) and \(x = -3i\). These solutions are called imaginary because they involve the imaginary unit \(i\).

• An imaginary root always comes in a pair with its conjugate because complex roots are symmetrical about the real axis. If \(3i\) is a root,\(-3i\) must also be a root.
• Imaginary roots do not correspond to a point on the real number line. Instead, they exist in a plane where the real part and the imaginary part define the position of the number.
• They are often used in mathematical calculations to ensure certain types of equations have solutions, thus broadening the applicability of algebraic methods.

Imaginary roots are an essential concept in higher mathematics, enabling the solution of equations that would otherwise be considered unsolvable.

Real Roots

Real roots are solutions to a polynomial equation where the roots are real numbers, meaning they do not involve imaginary numbers. For instance, in the equation \(x^2 = 9\), the solutions \(x = 3\) and \(x = -3\) are the real roots.

• Real roots are tangible and existing values on the number line that satisfy the equation.
• When plotting, real roots of an equation correspond to points where the graph intersects the x-axis.
• Real roots can be simple or repeated (multiple). In the given problem, each of the roots \(3\) and \(-3\) is simple, appearing exactly once.

Understanding real roots is crucial because they provide the visible and more intuitive solutions to polynomial equations, reflecting moments where a function crosses or touches the number line.