Problem 34 - Algebra 2

Question

Find the two real-number solutions of each equation. $$ x^{4}=1 $$

Step-by-Step Solution

Answer

The solutions to the given equation \(x^{4}=1\) are \(x = 1\) and \(x = -1\).

Step 1: Re-write the given equation

This question asks to find the real numbers for the equation \(x^{4}=1\). We can re-write this equation by considering positive and negative solutions of 1, giving two equivalent equations, \(x^{2} = 1\) and \(x^{2} = -1\) respectively.

Step 2: Solve the first equivalent equation

Now, solve the equation \(x^{2} = 1\). The solutions to this equation are \(x=1\) (since \(1^{2}=1\)) and \(x = -1\) (since \((-1)^{2}=1\)). So, \(x=1\) and \(x=-1\) are solutions.

Step 3: Solve the second equivalent equation

The next step is to solve the equation \(x^{2} = -1\). But this equation gives imaginary solutions, and not real solutions, for given \(x^{4}=1\). So there is no real solution from this equation.

Key Concepts

Polynomial Equations

Polynomial equations are fundamental in understanding algebra. They are equations that involve variables raised to whole-number powers, coupled with coefficients, and set equal to some value. For a polynomial equation like \(x^4 = 1\), the highest power of \(x\) is 4, making it a quartic equation.

Polynomials can be:

• Linear, like \(x + 1 = 0\)
• Quadratic, like \(x^2 - 4 = 0\)
• Cubic, like \(x^3 - 8 = 0\)
• Quartic, like our example \(x^4 = 1\)

To solve such equations, one often factors them or considers equivalent simpler equations. In the exercise, \(x^4 = 1\) can be split into \(x^2 = 1\) and \(x^2 = -1\), aiding in identification of solutions.

Real Numbers

Real numbers include all rational and irrational numbers. They comprise the number line and can be expressed as either terminating or non-terminating decimals. Real numbers include:

• Positive numbers (e.g., 2, 5.5)
• Negative numbers (e.g., -3, -7)
• Zero (0)

In the exercise, we seek the real-number solutions to \(x^4 = 1\). The real solutions emerge from solving \(x^2 = 1\), yielding \(x = 1\) and \(x = -1\). These numbers are both real because they exist on the number line and can be viewed as cuts or positions thereon.

However, \(x^2 = -1\) doesn't provide real solutions within the realm of real numbers, as multiplying the same real number by itself can't result in a negative.

Imaginary Numbers

Imaginary numbers rise from the necessity to solve polynomial equations that don't yield real solutions. The core unit of an imaginary number is \(i\), defined as \(i = \sqrt{-1}\). Imaginary numbers help solve equations like \(x^2 = -1\), where real numbers fail.

For instance:

• \(i^2 = -1\), clarifying why \(i\) lets us solve \(x^2 = -1\)
• Complex numbers include a real component and an imaginary one, such as \(3 + 4i\)

While finding solutions for \(x^4 = 1\), the equation \(x^2 = -1\) presents imaginary solutions, \(x = i\) and \(x = -i\), suitable in a broader mathematical context but not needed for real number solutions. Recognizing when imaginary numbers apply extends beyond basic algebra to more advanced topics.