Problem 21 - College Algebra

Question

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ x^{4}-8 x^{3}+64 x-105=0 ; 2-i $$

Step-by-Step Solution

Answer

The solution set to this polynomial equation includes the roots from step 2 and 4. Ensure to check the validity of each one to confirm the overall solution set.

Step 1: Confirm Given Root

Plug the given root \(2-i\) into the equation to confirm that it is indeed a root. A valid root will satisfy the equation \(x^{4}-8 x^{3}+64 x-105=0\).

Step 2: Find the Complex Conjugate

Every complex root \(a+bi\) of a polynomial with real coefficients has a complex conjugate \(a-bi\) as a root. So, the complex conjugate of \(2-i\) is \(2+i\). We now have two roots.

Step 3: Divide the Polynomial

Divide the polynomial \(x^{4}-8 x^{3}+64 x-105\) by \((x-(2-i))\) and \((x-(2+i))\). This will yield a quadratic equation.

Step 4: Solve Quadratic Equation

To find the other two roots, solve this resulting quadratic equation. There will be two real roots if the discriminant is non-negative, two complex roots if it is negative, or a single real root if it equals zero.

Step 5: Combine the Roots

The four roots of the original polynomial are the roots we found in step 2 and 4. These four roots represent the solution set to the equation.

Key Concepts

Complex Conjugate

When dealing with polynomial equations with real coefficients, complex roots often appear in conjugate pairs. This means, if we are given a complex number root such as \(2-i\), its complex conjugate would be \(2+i\). Complex conjugates are numbers with the same real part but opposite imaginary parts.

This property is crucial because it maintains the polynomial's real nature. Complex conjugates ensure that when multiplying the pairs, the imaginary parts cancel out, resulting in a real number. For instance:

• The complex number: \(a+bi\)
• Its complex conjugate: \(a-bi\)

This concept makes solving polynomial equations more straightforward when complex roots are involved. It also guarantees that polynomials such as the one in your original exercise are solvable using this fundamental principle.

Roots of a Polynomial

The roots of a polynomial are essentially the solutions to the polynomial equation, meaning they are the values of \(x\) that satisfy the equation \(P(x) = 0\). Finding roots is not just about obtaining numerical answers; it's an exploration of the behavior and properties of the polynomial function.

A polynomial of degree \(n\) will have exactly \(n\) roots, some of which may be repeated or complex. For example, if a fourth-degree polynomial like the one in your exercise has real coefficients, it will:

• Have up to 4 real roots.
• Or a combination of real roots and complex conjugate roots.

Understanding these roots helps in predicting the graph of the polynomial and recognizing the polynomial's end behavior.

Quadratic Equation

Quadratic equations are second-degree polynomial equations in the form \(ax^2 + bx + c = 0\). They are fundamental in algebra and have well-defined methods for finding their roots. The most common method is the quadratic formula, given by:

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

The expression under the square root, \(b^2 - 4ac\), is known as the discriminant. It determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If zero, there is exactly one real root (a repeated root).
• If negative, the roots are complex and occur as conjugates.

In your exercise, once the original polynomial is factored down by removing complex conjugate roots, you end up with a quadratic equation. Solving this using the quadratic formula gives the remaining roots of the original polynomial.

Factorization

Factorization in polynomials is the process of expressing the polynomial as a product of simpler polynomials which are called factors. This essentially breaks down a complex polynomial into parts that are easier to solve and understand.

The factorization process becomes crucial when trying to find roots of polynomials. By factorizing, you can identify common factors or roots, and further simplify solving. For the polynomial in your exercise, two of the factors were immediately identifiable using the complex conjugate theorem. Thus, after deriving the complex conjugate roots \((x - (2 - i))\) and \((x - (2 + i))\), you can factor these out of the polynomial.

After removing these known factors, the remaining polynomial is typically reduced to a quadratic, which can then be solved using standard techniques like factorization or utilizing the quadratic formula. This multi-step factorization helps uncover all roots step-by-step, making the problem more manageable and less prone to errors.