Question

Use a graphing utility to graph the polynomial functions \(p_{1}(x)=x^{3}-x+1\) and \(p_{2}(x)=x^{3}-x .\) How many zeros does each function have? Is there a cubic polynomial that has no zeros? Explain.

Short answer

The number of zeros in each function will vary and needs to be observed from the individual graphs. However, due to the Fundamental Theorem of Algebra, every cubic polynomial will have at least one real root, so it's impossible for a cubic polynomial to have no zeros.

Step-by-step solution

Graph \(p_{1}(x)=x^{3}-x+1\)

Use a graphing utility and plot \(p_{1}(x)=x^{3}-x+1\). Look for the points where the curve crosses the x-axis; these are the zeros of the function.

Find zeros of \(p_{1}(x)=x^{3}-x+1\)

From the graph, identify these points and note them down. If no points are crossing the x-axis then the function has no zeros.

Graph \(p_{2}(x)=x^{3}-x\)

Use the same graphing utility and plot the other function \(p_{2}(x)=x^{3}-x\). Again, look for the points where the curve crosses the x-axis; these are the zeros of the function.

Find zeros of \(p_{2}(x)=x^{3}-x\)

From the graph, identify these points and note them down. If no points are crossing the x-axis then the function has no zeros.

Analyze cubic polynomials

Consider whether a cubic polynomial can have no zeros. By the Fundamental Theorem of Algebra, every cubic polynomial has at least one real root.

Key concepts

Graphing Polynomial Functions

When it comes to understanding polynomial functions, one of the most insightful tools is graphing. A polynomial function of degree three, also known as a cubic polynomial, has the general form \(ax^3+bx^2+cx+d\), where \(a, b, c,\) and \(d\) are constants and \(a \eq 0\). Graphing these functions provides a visual representation of their behavior, including their curvature and importantly, where they intersect the x-axis, which are the points known as zeros.

For example, the given polynomial functions \(p_{1}(x)=x^3-x+1\) and \(p_{2}(x)=x^3-x\) can be graphed using a graphing utility. The graph of a cubic function typically features turns and intercepts with both axes. The intercepts with the x-axis are particularly useful as they indicate the values for which the function produces a result of zero. By closely examining the graph, students can visually comprehend how and where the function's curve crosses the x-axis. If the graph does not intersect the x-axis, then it may suggest complex roots that are not visible on the standard two-dimensional graph.

Finding Zeros of Polynomials

Identifying the zeros of a polynomial function—values of \(x\) for which the polynomial equals zero—is essential in many areas of mathematics. These zeros can represent real-life quantities, such as time or distance in physics problems, or profit and cost in economics. To find the zeros for the polynomials from our example, we need to solve the equations \(p_{1}(x)=x^3-x+1=0\) and \(p_{2}(x)=x^3-x=0\).

After graphing the functions, determining the zeros can be as straightforward as observing the x-coordinates where the graph intersects the x-axis. For \(p_{2}(x)\), it is evident that one of the zeros is x=0, because the function simplifies to \(x(x^2-1)\), revealing an intercept at the origin. For a polynomial like \(p_{1}(x)\), zeros may not be as easily factored, and thus, more advanced techniques like the Rational Root Theorem or numerical methods might be needed. Remember, however, that graphing only reveals the real zeros; polynomials may also have complex zeros that do not show up on a real number line graph.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is pivotal in polynomial function analysis. It states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This means that a cubic polynomial, which is of degree three, is guaranteed to have at least one real or complex zero. For students, this theorem assures that even when zeros cannot be identified visually on the graph, they still exist within the complex number system.

Consider our exercise, where we analyze the cubic polynomials \(p_{1}(x)\) and \(p_{2}(x)\). By applying the theorem, we understand that each function must have at least one zero. While \(p_{2}(x)\) visibly crosses the x-axis and reveals obvious zeros, there is a possibility that \(p_{1}(x)\) has a zero that does not intersect at a real-number coordinate on the x-axis. Therefore, through the Fundamental Theorem of Algebra, we know that if graphing does not show any real zeros, we are prompted to seek complex zeros. Furthermore, the answer to whether a cubic polynomial can have no zeros is a definitive no; it will always have one real zero at minimum due to this fundamental theorem.