Problem 16 - Algebra 2

Question

Solve each equation. State the number and type of roots. \(x^{5}-8 x^{3}+16 x=0\)

Step-by-Step Solution

Answer

The equation has five real roots: one simple root \(x = 0\), and two double roots \(x = 2\) and \(x = -2\).

Step 1: Factor out the Greatest Common Factor (GCF)

First, we notice that every term in the equation has a common factor of \(x\). We can factor \(x\) out of the equation: \(x(x^4 - 8x^2 + 16) = 0\). This gives us one solution: \(x = 0\).

Step 2: Solve the Quartic Polynomial

Now, we solve the remaining quartic equation: \(x^4 - 8x^2 + 16 = 0\). We can use substitution to simplify it. Let \(y = x^2\), then the equation becomes \(y^2 - 8y + 16 = 0\).

Step 3: Solve the Quadratic Equation

The equation \(y^2 - 8y + 16 = 0\) is a quadratic equation. We can solve it using the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -8\), and \(c = 16\).

Step 4: Calculate the Discriminant

The discriminant is \(b^2 - 4ac = (-8)^2 - 4(1)(16) = 64 - 64 = 0\). A discriminant of 0 means there is one real repeated root.

Step 5: Find the Roots of the Quadratic

As the discriminant is zero, the quadratic formula gives \(y = \frac{-(-8) \pm 0}{2(1)} = 4\). This is a repeated root. Since \(y = x^2\), we have \(x^2 = 4\).

Step 6: Solve for x

The solutions to \(x^2 = 4\) are \(x = \pm 2\). Thus, the remaining solutions are \(x = 2\) and \(x = -2\), each repeated twice due to the double root \(y = 4\).

Step 7: Identify the Number and Type of Roots

The original equation \(x^5 - 8x^3 + 16x = 0\) has five roots: \(x = 0\), \(x = 2\) (repeated twice), and \(x = -2\) (repeated twice). All roots are real.

Key Concepts

Roots of Polynomial

Finding the roots of a polynomial involves determining the values of the variable for which the polynomial evaluates to zero. In simpler terms, when the polynomial equals zero, these values are considered the roots. They represent the points where the polynomial crosses or touches the x-axis on a graph.

For example, in the polynomial equation \(x^{5} - 8x^{3} + 16x = 0\), by factoring out common terms and solving, we find five roots: \(x = 0\), \(x = 2\) (repeated twice), and \(x = -2\) (also repeated twice). Each root can tell us about the curve's behavior, such as points of intersection or tangency with the x-axis.

Understanding the roots is crucial as they help not only in sketching the polynomial graph but also in solving practical problems modeled by polynomials.

Quartic Equations

A quartic equation is a type of polynomial equation where the highest degree of the variable is four, typically taking the form \(ax^4 + bx^3 + cx^2 + dx + e = 0\).

Solving quartic equations can be intricate due to the number of possible solutions and their combinations. One common approach is to simplify the equation by substitution, converting it into a quadratic form, which is easier to handle.

• For instance, the original problem involves solving a reduced form of a quartic equation \(x^4 - 8x^2 + 16 = 0\).
• By letting \(y = x^2\), the equation simplifies to \(y^2 - 8y + 16 = 0\), a more familiar quadratic equation.

Such transformations are incredibly useful in handling quartic equations as they reduce complexity and help in applying known solutions like the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool used to find roots of quadratic equations, which have the form \(ax^2 + bx + c = 0\).

The formula is expressed as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It provides a straightforward method to solve any quadratic equation, even when factoring is difficult or impossible.

In the solved problem, the equation \(y^2 - 8y + 16 = 0\) uses the quadratic formula to find its roots. Here, \(a = 1\), \(b = -8\), and \(c = 16\). Using these values, we calculated the roots to be \(y = 4\), leading us back to the solutions for the original variable: \(x = 2\) and \(x = -2\). The quadratic formula is essential in many fields of mathematics, providing solutions that are both reliable and universally applicable.

Discriminant

The discriminant is a specific component of the quadratic formula \(b^2 - 4ac\). It helps us understand the nature and number of roots of a quadratic equation.

The discriminant can indicate:

• If it's positive, there are two distinct real roots.
• If it’s zero, there is one real repeated root (often called a double root).
• If negative, the roots are complex or imaginary, occurring in conjugate pairs.

In our problem, the discriminant calculated for \(y^2 - 8y + 16 = 0\) is zero, suggesting a repeated root. That means the graph of the equation touches the x-axis at one distinct point. Understanding the discriminant gives us insight into the behavior of the equation's solutions without needing to solve it completely right away.

Factorization

Factorization is the process of decomposing a polynomial into a product of simpler polynomials which, when multiplied, give the original polynomial.

This technique simplifies the process of finding roots and solving the equation. In the provided exercise, the first step involved factoring out the Greatest Common Factor (GCF), \(x\), from the equation \(x^5 - 8x^3 + 16x = 0\).

• This reduced the equation to \(x(x^4 - 8x^2 + 16) = 0\).
• Factorization also helps in showing the multiplicity of roots, such as repeated roots.

By breaking down complex expressions, factorization reveals insights into the structure of the polynomial and allows for an easier path to discovering solutions. It's a foundational tool that bridges basic algebra with more advanced mathematical problem-solving.