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Magazine Chapter 4: Problem 37
Solve the polynomial equation. $$ x^{4}=x^{3}-4 x^{2} $$
Short Answer
Expert verified
The solutions are \( x = 0 \), \( x = \frac{1 + i\sqrt{15}}{2} \), and \( x = \frac{1 - i\sqrt{15}}{2} \).
Step by step solution
01
Set Equation to Zero
Start by moving all terms to one side of the equation to set it to zero. The original equation is \( x^4 = x^3 - 4x^2 \). Subtract \( x^3 \) and add \( 4x^2 \) to both sides to get: \( x^4 - x^3 + 4x^2 = 0 \).
02
Factor Out the Common Term
Identify a common factor in the equation. All terms contain at least \( x^2 \). Factor \( x^2 \) from the equation: \( x^2(x^2 - x + 4) = 0 \). This gives a product of terms equal to zero.
03
Solve Each Factor Separately
For the factored equation \( x^2(x^2 - x + 4) = 0 \), solve the product by zero property: \[ x^2 = 0 \] and \[ x^2 - x + 4 = 0 \].
04
Solve for x from x^2 = 0
Solve \( x^2 = 0 \). When a square equals zero, the solution is \( x = 0 \).
05
Solve the Quadratic x^2 - x + 4 = 0
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve the quadratic \( x^2 - x + 4 = 0 \), with \( a = 1, b = -1, c = 4 \).
06
Calculate the Discriminant
First, calculate the discriminant: \( b^2 - 4ac = (-1)^2 - 4 \times 1 \times 4 = 1 - 16 = -15 \). The discriminant is negative, indicating complex roots.
07
Find Complex Roots
Compute the complex roots using the quadratic formula; \( x = \frac{-(-1) \pm \sqrt{-15}}{2 \cdot 1} \) simplifies to \( x = \frac{1 \pm i\sqrt{15}}{2} \). The solutions are \( x = \frac{1 + i\sqrt{15}}{2} \) and \( x = \frac{1 - i\sqrt{15}}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Factoring polynomials is a crucial skill in algebra that simplifies equations and exposes potential solutions. Begin by examining the polynomial and identifying any common factors across all terms. In the example equation \(x^4 - x^3 + 4x^2 = 0\), observe that each term includes at least \(x^2\) as a factor. Extracting \(x^2\) results in a simpler form: \(x^2(x^2 - x + 4) = 0\). This separation into factors allows us to apply the zero product property, setting each factor to zero individually.
- The zero product property states that if a product of terms equals zero, at least one of the terms must also be zero.
- This principle enables us to solve \(x^2 = 0\) and \(x^2 - x + 4 = 0\) separately.
Complex Roots
Complex roots often emerge when solving polynomial equations with a negative discriminant. Such a discriminant tells us that real number solutions do not exist, leading us into the realm of complex numbers. In the equation \(x^2 - x + 4 = 0\), we find the discriminant \(b^2 - 4ac = -15\). As this value is negative, the quadratic equation possesses roots that include imaginary numbers.
- Complex numbers consist of a real part and an imaginary part, expressed as \(a + bi\), where \(i\) is the square root of \(-1\).
- Imaginary roots appear in pairs, thus ensuring polynomial equations have a balanced set of solutions.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). Applicable to any quadratic equation, the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) provides direct access to its roots. Let's break down this tool in our provided equation:
- Identify \(a = 1\), \(b = -1\), and \(c = 4\).
- Calculate the discriminant \(b^2 - 4ac\) to predict the nature of the roots.
- A negative discriminant, as seen here, indicates the presence of complex roots.
