Problem 65 - Algebra and Trigonometry

Question

By the Zeros Theorem, every \(n\) th-degree polynomial equation has exactly \(n\) solutions (including possibly some that are repeated). Some of these may be real and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has. (a) \(x^{4}-2 x^{3}-11 x^{2}+12 x=0\) (b) \(x^{4}-2 x^{3}-11 x^{2}+12 x-5=0\) (c) \(x^{4}-2 x^{3}-11 x^{2}+12 x+40=0\)

Step-by-Step Solution

Answer

(a) 3 real, 1 imaginary; (b) 1 real, 3 imaginary; (c) 0 real, 4 imaginary.

Step 1: Analyze the Given Polynomial

The polynomial equation given, \(x^4 - 2x^3 - 11x^2 + 12x = 0\), is a degree 4 polynomial. According to the Zeros Theorem, it will have 4 roots in total, which may be real or imaginary, and some might be repeated.

Step 2: Find Real Solutions Using Graph

Use a graphing tool to plot the graph of \(x^4 - 2x^3 - 11x^2 + 12x\). Observe where the graph crosses or touches the x-axis. These intersections are the real roots.

Step 3: Analyze First Polynomial Graph

Upon graphing, the expression \(x^4 - 2x^3 - 11x^2 + 12x\) appears to intersect the x-axis at three distinct points, suggesting three real roots.

Step 4: Determine Imaginary Roots for Part (a)

Since there are three real roots and it is a degree 4 polynomial, there must be one imaginary root (considering multiplicity and complex roots often come in pairs).

Step 5: Analyze the Second Polynomial \(x^4-2x^3-11x^2+12x-5=0\)

Graph the polynomial \(x^4 - 2x^3 - 11x^2 + 12x - 5\). Check for intersections with the x-axis to find real roots.

Step 6: Interpret Graph for Part (b)

The graph for \(x^4 - 2x^3 - 11x^2 + 12x - 5\) shows one intersection with the x-axis, indicating one real root.

Step 7: Determine Imaginary Solutions for Part (b)

With one real root and a total degree of 4, there must be three imaginary roots.

Step 8: Graph the Third Polynomial \(x^4-2x^3-11x^2+12x+40=0\)

Plot the polynomial on a graphing device to see where it touches the x-axis.

Step 9: Analyze Graph for Part (c)

The graph for \(x^4 - 2x^3 - 11x^2 + 12x + 40\) does not intersect the x-axis, suggesting no real roots.

Step 10: Determine Imaginary Roots for Part (c)

Since there are no real roots and it is a degree 4 polynomial, all four roots must be imaginary.

Key Concepts

Real Roots

Real roots of a polynomial are the x-values where the polynomial actually crosses or touches the x-axis on a graph. These are also known as the solutions or zeroes of the polynomial that are real numbers. When you graph a polynomial, wherever it touches or crosses the x-axis, you will find your real roots. This can help you visually confirm the number of real solutions an equation has. If the polynomial equation has repeating real roots, the graph may just touch the x-axis without crossing it, which is also known as a root of multiplicity. Run a graphing tool for the polynomial and observe to estimate the real roots. Remember, real roots are simply the x-values at those intersections that lie on the real number line.

Imaginary Roots

Imaginary roots occur when a polynomial does not intersect the x-axis at certain points. Instead of touching or crossing the x-axis, the graph remains either entirely above or below it between roots or on intervals where it should have crossed. Imaginary roots typically come in conjugate pairs in real coefficients polynomials, making them appear complex conjugates when expressed mathematically. This means that if \(a + bi\) is a root, \(a - bi\) is also a root. In a polynomial of degree 4, for instance, if you identify 2 real roots, the remaining roots are imaginary and likely will form these conjugate pairs. Imaginary roots are crucial because they preserve the polynomial's degree even when there are fewer real solutions than expected.

Zeros Theorem

The Zeros Theorem is a vital concept that states every nth-degree polynomial equation has exactly n solutions or roots. These roots could be real, imaginary, or repeated, but their total count including multiplicity will equal the degree of the polynomial. This theorem doesn’t just provide a number but also guides you in expecting a mix of possible solutions. For a polynomial like \(x^4 - 2x^3 - 11x^2 + 12x\), which is of degree 4, expect four roots in total. However, with some real and complex roots, some might repeat at identical x-values, residue imaginary, or involve multiplicity that doesn't directly show separate intersections on a graph.

Graphing Polynomials

Graphing polynomials can be a powerful tool for understanding their roots, behavior, and characteristics. By plotting a polynomial on a graph, one can easily see where and how many times it crosses the x-axis, signifying the real roots. Start by understanding polynomial behavior based on its highest degree, which gives rise to the general shape of the curve. A graphing calculator or software can come handy to analyze even complex polynomials. Pay attention to multiple roots where the graph just touches the x-axis or remains above or below signaling imaginary roots. For polynomials like \(x^4 - 2x^3 - 11x^2 + 12x\), visual observation via graphing will reveal intersections and the absence thereof, giving a clear idea of both real and non-real roots.