Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, use Descartes' Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$ f(x)=2 x^{4}-5 x^{3}-14 x^{2}+20 x+8 $$

Step-by-Step Solution

Answer

The polynomial has 2 positive roots and 0 negative roots.

Step 1: Count Sign Changes for Positive Roots

To determine the number of positive real roots, analyze the polynomial as it is: \( f(x) = 2x^4 - 5x^3 - 14x^2 + 20x + 8 \).Observe sign changes between consecutive non-zero coefficients. 1. From \(+2\) to \(-5\) — 1st sign change.2. From \(-5\) to \(-14\) — no change.3. From \(-14\) to \(+20\) — 2nd sign change.4. From \(+20\) to \(+8\) — no change.There are 2 sign changes, hence there can be 2 or 0 positive roots.

Step 2: Count Sign Changes for Negative Roots

To find the number of negative real roots, evaluate \( f(-x) \):\( f(-x) = 2(-x)^4 - 5(-x)^3 - 14(-x)^2 + 20(-x) + 8 \= 2x^4 + 5x^3 - 14x^2 - 20x + 8 \).Now count the sign changes:1. From \(+2\) to \(+5\) — no change.2. From \(+5\) to \(-14\) — 1st sign change.3. From \(-14\) to \(-20\) — no change.4. From \(-20\) to \(+8\) — 2nd sign change.There are 2 sign changes, which means there can be 2 or 0 negative roots.

Step 3: Analyze Using the Graph

When looking at the graph of the function, we see that it has 2 positive roots and 0 negative roots. This confirms the possible numbers of roots we found using Descartes' Rule of Signs.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's a formula made up of various powers of a variable, like \(x\), each multiplied by a given number, called the coefficient. For the polynomial \(f(x) = 2x^4 - 5x^3 - 14x^2 + 20x + 8\):

• The highest power of \(x\) is 4, making it a fourth-degree polynomial.
• Each term is a combination of a coefficient and a power of \(x\), like the term \(-5x^3\).
• The behavior of the polynomial function can often be determined by its degree and the signs of its coefficients.

Understanding polynomial functions is key to solving equations and analyzing their graphical representations.
When encountering polynomial functions, always observe their degree, leading coefficients, and the terms in between. These details predict how the function will perform across different values of \(x\).

Roots of Equations

Roots of an equation are the solutions where the equation equals zero, meaning the values of \(x\) that make the function \(f(x) = 0\). Using Descartes' Rule of Signs, we anticipate the number of possible positive and negative roots by counting the sign changes in the coefficients of the polynomial:

• By counting sign changes in \(f(x)\), we determine possible positive roots.
• To check for negative roots, evaluate \(f(-x)\) and count the sign changes.

For our function, by evaluating \(f(x)\), we find 2 or 0 positive roots. With \(f(-x)\), we also predict 2 or 0 negative roots. These potential outcomes highlight that not every root predicted by the rule will exist physically; it only sets boundaries on possibilities. Roots can often be confirmed through graphical analysis or algebraic methods such as synthetic division or factoring.

Graphical Analysis

Graphical analysis involves plotting the polynomial function on a graph to visually inspect its behavior. This is particularly useful in verifying the roots and sign behavior predicted by Descartes' Rule of Signs. When graphed, the polynomial function \(f(x) = 2x^4 - 5x^3 - 14x^2 + 20x + 8\) exhibits specific characteristics:

• Intercepts with the x-axis indicate the roots of the polynomial.
• The shape and direction of the curve help deduce local minima and maxima, and whether the function has more real roots.
• The confirmed presence of 2 positive roots aligns with our sign analysis, showing accurate practical predictions from theoretical analysis.

By graphing the polynomial, students can physically see the fall and rise at different intervals, associating these visual cues with algebraic calculations. Graphical analysis serves as a complete picture, making abstract root calculations clearer by grounding them in visual evidence.