Problem 40 - College Algebra with Modeling and Visualization

Question

Solve the polynomial equation. $$ x^{4}+2 x^{2}=x^{3} $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \), \( x = \frac{1 + i\sqrt{7}}{2} \), and \( x = \frac{1 - i\sqrt{7}}{2} \).

Step 1: Move All Terms to One Side

Begin by rewriting the equation so that all terms are on one side of the equation. Start with the original equation: \[ x^{4} + 2x^{2} = x^{3} \] Subtract \( x^{3} \) from both sides to obtain: \[ x^{4} - x^{3} + 2x^{2} = 0 \]

Step 2: Factor Out Common Factors

Look for common factors in the polynomial. Notice that \( x^{2} \) is a common factor in all terms: \[ x^{2}(x^{2} - x + 2) = 0 \] This simplifies to a product of terms equal to zero.

Step 3: Solve Each Factor Separately

Set each factor equal to zero and solve: 1. For the factor \( x^{2} = 0 \), solve for \( x \): \[ x = 0 \] 2. For the factor \( x^{2} - x + 2 = 0 \), use the quadratic formula to solve it. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -1 \), \( c = 2 \). Thus, \[ x = \frac{1 \pm \sqrt{(-1)^{2} - 4 \times 1 \times 2}}{2 \times 1} \] \[ x = \frac{1 \pm \sqrt{1 - 8}}{2} \] \[ x = \frac{1 \pm \sqrt{-7}}{2} \] The expression under the square root is negative, implying non-real (complex) solutions:\[ x = \frac{1 \pm i\sqrt{7}}{2} \]

Step 4: Determine All Solutions

Compile the real and complex solutions from the previous steps: - Real solution: \( x = 0 \)- Complex solutions from the quadratic formula: \( x = \frac{1 + i\sqrt{7}}{2} \) and \( x = \frac{1 - i\sqrt{7}}{2} \)

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves breaking down a complex expression into simpler parts that can be multiplied together to yield the original polynomial.
To factor a polynomial effectively, you need to recognize common factors that can be extracted from all terms.
In the given exercise, we started with the equation:

• \( x^4 + 2x^2 = x^3 \)

After moving all terms to one side, we rearranged it to:

• \( x^4 - x^3 + 2x^2 = 0 \)

The key is to identify the greatest common factor in the polynomial terms. Here, each term has \( x^2 \) as a factor, which can be factored out:

• \( x^2(x^2 - x + 2) = 0 \)

By factoring out \( x^2 \), we simplify the equation significantly. Each factor can now be analyzed separately to find solutions for \( x \). This process highlights how factoring transforms a complex equation into more manageable components.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). When factoring is difficult or impossible, the quadratic formula provides a systematic way to find solutions.
The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In the original exercise, we applied the quadratic formula to the factor \( x^2 - x + 2 = 0 \), where:

• \( a = 1 \)
• \( b = -1 \)
• \( c = 2 \)

Plugging these values into the formula:

• \( x = \frac{1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \)

We find:

• \( x = \frac{1 \pm \sqrt{-7}}{2} \)

This yields solutions involving square roots of a negative number, leading us into the realm of complex numbers. The quadratic formula thus not only helps find real solutions but also guides us in identifying complex solutions when necessary.

Complex Solutions

While solving polynomial equations, we may encounter solutions that involve square roots of negative numbers. These are called complex solutions.
Complex solutions arise because the square root of a negative number is not a real number. Instead, it includes the imaginary unit \( i \), where \( i^2 = -1 \).
From our exercise, when using the quadratic formula for \( x^2 - x + 2 = 0 \), we encountered:

• \( x = \frac{1 \pm i\sqrt{7}}{2} \)

This is because the term under the square root (the discriminant) was negative: \( -7 \). Complex numbers have the form \( a + bi \), where \( a \) and \( b \) are real numbers.
Our solutions break down to:

• \( x = \frac{1 + i\sqrt{7}}{2} \)
• and \( x = \frac{1 - i\sqrt{7}}{2} \)

Understanding complex solutions is crucial when real numbers do not suffice for finding roots of a polynomial. They expand our capability to address more sophisticated mathematical problems.