Problem 41 - Algebra for College Students

Question

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{4}+x^{2}+1=0$$

Step-by-Step Solution

Answer

The given equation has no real roots. The complex roots present in conjugate pairs

Step 1: Identify the type of equation

The given equation is a polynomial equation of degree 4:

Step 2: Analyze the discriminant

For polynomial equations of degree higher than 2, we analyze based on properties of the polynomial. For a quadratic equation of the form , there are criteria involving that can tell us about the nature of the roots.

Step 2: Check for real coefficients symmetry

Consider the symmetry of the polynomial. If all the coefficients of the polynomial (here, ) are real, then complex roots, if any, of the polynomial occur in conjugate pairs.

Step 3: Analyze via substitution

Let's substitute in the given cubic polynomial. This converts it into: After the substitution,.

Step 4: Discuss possible nature of given in

Analyzing how the graph behaves, the roots of the polynomial equation can be characterized happening: .

Key Concepts

Degree of Polynomial

A polynomial's degree is the highest power of the variable in the polynomial equation. This helps in understanding the behavior and possible number of roots the equation can have.

In our case, the given equation is \(x^{4} + x^{2} + 1 = 0\). The highest power of \(x\) is 4, so the degree of polynomial is 4. This tells us that there are 4 roots (real or complex), as a polynomial of degree \(n\) has exactly \(n\) roots in the complex number system.

Knowing the degree helps to anticipate how many times we need to factorize or apply the quadratic formula to find the roots.

Complex Roots

Polynomials with real coefficients often have roots that are not just real numbers. If any of the coefficients are real, complex roots will appear in conjugate pairs. This means if \(a + bi\) is a root, then \(a - bi\) will also be a root.

For our polynomial \(x^{4} + x^{2} + 1 = 0\), all coefficients are real. Therefore, if there are any complex roots, they must come in pairs. This ensures that the polynomial remains valid with real coefficients.

When checking for complex roots, we transform the equation using various methods, like substitution, to simplify and analyze it further.

Substitution Method

To simplify polynomials, we often use substitution. In this method, a variable is replaced with another variable that converts the polynomial into a simpler form.

For the equation \(x^{4} + x^{2} + 1 = 0\), we can let \(y = x^{2}\). Substituting \(y\) transforms the polynomial into \(y^{2} + y + 1 = 0\).

This substitution converts the original quartic equation into a quadratic one, making it easier to handle. Once solved, the substitution is reversed to find the original variable's roots.

Polynomial Coefficients

The coefficients of a polynomial are the numbers that multiply the variables. They influence the shape of the graph and the nature of the roots.

In our equation, the coefficients are 1 (for \(x^{4}\)), 1 (for \(x^{2}\)), and 1 (constant term).

Understanding the role of coefficients is critical as they determine the equation's symmetry and root behavior. With all real coefficients, we can predict that complex roots, if they exist, will be in conjugate pairs, preserving the overall property of the polynomial having real numbers as coefficients.