Problem 65 - College Algebra with Corequisite Support

Question

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=16 x^{4}-24 x^{3}+x^{2}-15 x+25 $$

Step-by-Step Solution

Answer

Graph the function to find potential zeros, then verify them using synthetic division.

Step 1: Graph the Polynomial Function

Use a graphing calculator or graphing software to plot the polynomial function \( f(x) = 16x^4 - 24x^3 + x^2 - 15x + 25 \). Analyze the graph to determine where the function crosses the x-axis. These points will be your potential rational zeros.

Step 2: Identify Potential Rational Zeros

From the graph, note the x-values where the polynomial intersects the x-axis. These intersection points represent the rational zeros of the function. Record these x-values for verification.

Step 3: Use Synthetic Division to Verify Zeros

For each potential rational zero identified, use synthetic division to verify if substitution of this zero results in a remainder of zero. If the remainder is zero, the x-value is a confirmed rational zero of the polynomial function.

Step 4: Confirm Rational Zeros

After applying synthetic division, confirm which values are truly rational zeros. Ensure those values satisfy the equation \( f(x) = 0 \). List these confirmed rational zeros.

Key Concepts

Rational Zeros

The rational zeros of a polynomial are the x-values where the polynomial equals zero. They are called "zeros" because they make the function value equal to zero when substituted into the equation. To find them, first, graph the polynomial to see approximately where it crosses the x-axis. These crossings might be potential rational zeros, though not every crossing guarantees a rational zero.

Once you identify these crossings, you can use a process of elimination to test which values are actually zeros. The Rational Root Theorem can be helpful here as it provides potential candidates for rational zeros based on the coefficients of the polynomial. However, the graph gives you the initial visual indication of where these zeros might lie. Look for clean intersections, which hint at rational numbers. These potential zeros then need to be verified analytically.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form \(x - c\). It’s especially useful in confirming whether a number is a rational zero of a polynomial function. If, after performing synthetic division, you end with a remainder of zero, you’ve found a rational zero.

The steps for synthetic division are straightforward:

• Write down the coefficients of the polynomial.
• Place the potential zero you want to test outside the division bracket.
• Bring down the first coefficient as it is.
• Multiply this number by the potential zero and add it to the next coefficient. Write the result below.
• Continue this process until you’ve worked through all coefficients.
• If the last number is zero, the potential zero is indeed a rational zero.

Using synthetic division is less cumbersome than polynomial long division and is practical for quickly testing possible zeros noted from the graph.

Graphing Calculator Tools

A graphing calculator tool is incredibly helpful when dealing with polynomials. It visualizes where the function crosses the x-axis, which can be less apparent through calculation alone. These tools allow you to quickly sketch the graph, zoom in and out to refine the view, and trace the curve to find intersection points more accurately.

When you graph a polynomial like \( f(x) = 16x^4 - 24x^3 + x^2 - 15x + 25 \), the graph may show twisted curves and multiple crossings. Tracing the polynomial directly on the calculator helps to identify the nearest potential zeros. Some calculators even provide built-in functions to find zeros, which instantly solve for where the function equals zero.

Beyond simple visualization, using graphing calculators saves time and reduces human error in calculations. They provide immediate insights which aid in further analysis, especially for polynomials with higher degrees, as graphing by hand would be cumbersome and potentially inaccurate.