Problem 35 - Algebra for College Students

Question

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{3}+3 x^{2}+5 x+7=0$$

Step-by-Step Solution

Answer

The equation has three roots that could be either three real roots or one real root and two complex conjugates.

Step 1: Identify the Degree of the Polynomial

Notice that the given equation is a cubic polynomial: x^{3}+3x^{2}+5x+7=0. Since the highest power of x is 3, this is a third-degree polynomial. Third-degree polynomials have exactly three roots, which could be real or complex numbers.

Step 2: Evaluate Using the Fundamental Theorem of Algebra

According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots in the complex number system. Hence, the polynomial x^{3}+3x^{2}+5x+7=0 has exactly three roots.

Step 3: Determine the Nature of the Roots

The roots of a cubic equation can be all real, or one real and two complex conjugates. Examining the discriminant of a cubic equation can provide insights; however, without solving, note the polynomial could have: - Three distinct real roots. - One real root and two non-real complex conjugate roots.

Step 4: Summary of Possible Root Types

Summarizing the possibilities: - The equation can have three real roots. - Alternatively, it can have one real root and two complex roots.

Key Concepts

Cubic Polynomial

A cubic polynomial is a polynomial of degree three. In our equation,
\(x^{3} + 3x^{2} + 5x + 7 = 0\), the highest exponent of the variable \(x\) is 3. This characteristic means we are working with a cubic polynomial.

Cubic polynomials can have up to three roots. These roots can be real or complex. Understanding the nature of the polynomial is the first step in exploring its roots.

Remember, the general form of a cubic polynomial is: \(ax^{3} + bx^{2} + cx + d = 0\). Here, \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a key principle when dealing with polynomial equations. It states that every non-zero polynomial equation of degree \(n\) has exactly \(n\) roots in the complex number system.

For our cubic polynomial \(x^3 + 3x^2 + 5x + 7 = 0\), this theorem implies there are exactly three roots. These roots might be real or complex numbers.

The theorem guarantees the existence of these roots but does not provide their exact values or nature. Hence, we know there are three roots but need further analysis to understand their specific types.

Discriminant of Cubic Equations

The discriminant of a cubic equation helps determine the nature of its roots. For a general cubic equation \(ax^3 + bx^2 + cx + d = 0\), the discriminant \( \text{Δ} \) is a specific value calculated using \(a\), \(b\), \(c\), and \(d\).

The discriminant provides insights into whether the roots are real or complex:

• If \( \text{Δ} > 0 \), the equation has three distinct real roots.
• If \( \text{Δ} = 0 \), the equation has a multiple root, and all its roots are real.
• If \( \text{Δ} < 0 \), the equation has one real root and two non-real complex conjugate roots.

Understanding the discriminant's role is crucial for predicting root nature without solving the equation.

Real and Complex Roots

Polynomials can have real or complex roots. Real roots are values of \(x\) that satisfy the equation and lie on the real number line.

Complex roots involve imaginary numbers and do not lie on the real number line. They can be expressed in the form \(a + bi\), where \(i\) is the imaginary unit.

Cubic polynomials can have different combinations of real and complex roots:

• All three roots can be real.
• There can be one real root and two complex roots that are conjugates of each other.

Understanding this distinction helps in analyzing the root structure of the polynomial.