Problem 415 - Precalculus

Question

For the following exercises, find all complex solutions (real and non-real). $$ x^{3}-8 x^{2}+25 x-26=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2, 3 + 2i, 3 - 2i\).

Step 1: Check for Rational Roots

Begin by applying the Rational Root Theorem which suggests that any rational root, expressed as \(\frac{p}{q}\), will have \(p\) as a factor of the constant term (-26) and \(q\) as a factor of the leading coefficient (1). The possible rational roots are \(\pm 1, \pm 2, \pm 13, \pm 26\). Testing these values in the polynomial to check if any yields zero will identify a root.

Step 2: Test Potential Rational Roots

Evaluate the polynomial \(x^3 - 8x^2 + 25x - 26\) for each possible root: \(x = 1\), yield 1; \(x = 2\), yields 0. So, \(x = 2\) is a root of the polynomial.

Step 3: Factor out the Known Root

Since \(x = 2\) is a root, \(x - 2\) is a factor. Use polynomial division to divide the polynomial \(x^3 - 8x^2 + 25x - 26\) by \(x - 2\). The division yields the quotient \(x^2 - 6x + 13\).

Step 4: Solve the Quadratic Equation

The quotient from the division is \(x^2 - 6x + 13 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -6\), and \(c = 13\). Calculate the discriminant as \((-6)^2 - 4\times1\times13 = 36 - 52 = -16\), indicating complex roots.

Step 5: Determine the Complex Roots

Given the negative discriminant, the roots are complex. Calculate using the quadratic formula: \(x = \frac{6 \pm \sqrt{-16}}{2} = \frac{6 \pm 4i}{2} = 3 \pm 2i\). Hence, the two complex solutions are \(3 + 2i\) and \(3 - 2i\).

Step 6: Conclude with all Solutions

Combining all solutions: the polynomial has one real solution \(x = 2\) and two complex solutions \(3 + 2i\) and \(3 - 2i\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool in determining the potential rational roots of a polynomial equation. This theorem states that any rational solution of a polynomial, expressed as \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

In practice, this means:

• Identifying the constant term. For our polynomial \( x^3 - 8x^2 + 25x - 26 \), it's \(-26\).
• Identifying the leading coefficient. In the given polynomial, it's 1.
• Listing the factors of \(-26\) which are \( \pm 1, \pm 2, \pm 13, \pm 26 \).
• The factors of 1 are only \( \pm 1 \).

Therefore, the possible rational roots are the factors of \(-26\) divided by the factors of 1, which remain \( \pm 1, \pm 2, \pm 13, \pm 26 \). Each of these potential roots is tested in the polynomial to determine if it results in a zero value.

Polynomial Division

Polynomial division is a technique used to simplify polynomials, just like regular division simplifies numbers. When you discover a root of a polynomial, you also find a corresponding factor.

With our polynomial \( x^3 - 8x^2 + 25x - 26 \), we found that \( x = 2 \) is a root. This means \( x - 2 \) is a factor of the polynomial. To simplify the polynomial, we divide it by \( x - 2 \).

Performing polynomial division, we write the polynomial in long division format and divide the first term by the factor's leading term, \( x \). Subtract, bring down the next terms, and repeat this process.

Through this division, the original polynomial simplifies to \( x^2 - 6x + 13 \) with no remainder, confirming successful division and simplifying our polynomial for further solutions.

Quadratic Formula

The quadratic formula is a method used to find the roots of a quadratic equation, which is any equation of the form \( ax^2 + bx + c = 0 \). The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This allows you to calculate the roots directly by substituting the coefficients \( a \), \( b \), and \( c \).

In \( x^2 - 6x + 13 = 0 \), use:

• \( a = 1 \)
• \( b = -6 \)
• \( c = 13 \)

First, calculate the discriminant \( b^2 - 4ac \). Here, it's \((-6)^2 - 4 \times 1 \times 13 = -16 \). The negative value indicates the roots are complex. Plug these into the quadratic formula to find:

• \( x = 3 + 2i \)
• \( x = 3 - 2i \)

These are the complex roots of the equation.

Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are expressed in the form \( a + bi \), where:\

• \( a \) is the real part.
• \( bi \) is the imaginary part, with \( i \) being the square root of \(-1\).

In the quadratic equation from our exercise, the imaginary unit \( i \) arises from the square root of a negative number in the discriminant.

Complex numbers are useful in capturing solutions that are not possible with just real numbers. In our equation \( x^2 - 6x + 13 = 0 \), the solutions are \( 3 + 2i \) and \( 3 - 2i \). Here, \( 3 \) is the real part and \( 2i \) is the imaginary part.

Understanding these concepts helps in visualizing that numbers exist beyond the regular number line, involving both real and non-real dimensions.