Problem 57 - A Graphical Approach to College Algebra

Question

For each polynomial function, (a) list all possible rational zeros, (b) use a graph to eliminate some of the possible zeros listed in part (a), (c) find all rational zeros, and (d) factor \(P(x)\). $$P(x)=6 x^{3}+17 x^{2}-31 x-12$$

Step-by-Step Solution

Answer

The rational zeros are \(x = -3, -1/3, 2\). The polynomial factors as \(P(x) = (x + 3)(x - 2)(3x + 1)\).

Step 1: List Possible Rational Zeros

To find possible rational zeros of the polynomial \(P(x) = 6x^3 + 17x^2 - 31x - 12\), we use the Rational Root Theorem. This theorem states that any rational zero, \(p/q\), must have \(p\) as a factor of the constant term (-12) and \(q\) as a factor of the leading coefficient (6). Thus, the possible rational zeros are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm 1/2, \pm 1/3, \pm 1/6, \pm 2/3, \pm 3/2, \pm 3/6 \).

Step 2: Graph to Eliminate Options

Next, graph the function to identify points where the polynomial crosses the x-axis. This provides visual insight into which of the listed rational zeros are likely candidates. You can use graphing tools to do this. Look at the graph and note any points exactly on the x-axis, which correspond to the rational zeros.

Step 3: Confirm Rational Zeros

Evaluate the polynomial at each of the potential rational zeros from Step 1. Substitute these values into \(P(x)\) to determine if \(P(x) = 0\). Begin by testing with obvious integers seen on the graph. If \(x = a\) results in \(P(a) = 0\), then \(x = a\) is a root.

Step 4: Divide Polynomial by (x-a)

Once a root \(x = a\) is found, use synthetic division to divide \(P(x)\) by \((x-a)\), simplifying the polynomial. Continue this process using any additional rational zeros found in Step 3. Complete the division to obtain a new lower-degree polynomial.

Step 5: Factor Fully

With the lower-degree polynomial obtained in Step 4, repeat steps on any remaining feasible rational roots to fully factor. Combine each linear factor from successful divisions to express \(P(x)\) fully factored.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool when working with polynomials. It helps us determine the possible rational zeros of a polynomial by focusing on the factors of its constant term and leading coefficient.

• To find rational zeros, consider a polynomial of the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0\).
• The theorem states that any potential rational root \(\frac{p}{q}\) must have \(p\) as a divisor of \(a_0\) (the constant term) and \(q\) as a divisor of \(a_n\) (the leading coefficient).

For instance, if we have \(P(x) = 6x^3 + 17x^2 - 31x - 12\), the possible rational zeros are derived from the factors of \(-12\) (i.e., \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\)) and the factors of \(6\) (i.e., \(\pm 1, \pm 2, \pm 3, \pm 6\)). Combine these to get all potential rational zero candidates.

Synthetic Division

Synthetic division is a simplified form of polynomial division, especially useful when dividing by linear forms like \(x - a\). It provides a quick way to evaluate potential roots. Here's a basic rundown of how synthetic division works:

• Write down the coefficients of the polynomial.
• Place the candidate root (from Rational Root Theorem testing) to the side.
• Carry the first coefficient down, multiply it by the candidate root, and add to the next coefficient. Repeat until all are processed.
• If the last number (the remainder) is zero, then \(x = a\) is indeed a root. The other numbers give the coefficients of the reduced polynomial.

This process not only confirms whether a candidate is a root but also breaks the polynomial into simpler, lower-degree polynomials for further investigation.

Factoring Polynomials

Factoring polynomials is about expressing a polynomial as a product of its roots or simpler polynomials. After using tools like the Rational Root Theorem and synthetic division, we can often completely break down a polynomial into polynomial factors. Here’s how to approach this:

• First, confirm any rational root and use it to divide the polynomial using synthetic division.
• The quotient retrieves a lower-degree polynomial that can often be factored further.
• Repeat the process on the new polynomial until obtaining linear factors or irreducible polynomials.

Ultimately, the goal is to express the polynomial as a multiplication of these simpler expressions, showing clearly all values for which the polynomial equals zero.

Graphing Polynomials

Graphing is a powerful visual tool that complements algebraic methods while working with polynomials. When you plot a polynomial function, such as \(P(x) = 6x^3 + 17x^2 - 31x - 12\), you gain insight into its behavior:

• Understand the general shape of the curve (e.g., ascending or descending based on leading coefficient).
• Identify where the polynomial crosses the x-axis — these are the roots.
• Observe turning points, which are places the graph changes direction — valuable for understanding maximum and minimum values.

By graphing, certain potential rational zeros can be visually eliminated if they do not occur where the graph crosses the x-axis. It helps narrow down the candidates from the Rational Root Theorem, avoiding unnecessary calculations.