Problem 57 - A Graphical Approach to Precalculus with Limits

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

Rational zeros are \(-3, \frac{1}{2}, 4;\) factor: \(P(x) = 6(x + 3)(2x - 1)(x - 4)\).

Step 1: Determine Possible Rational Zeros

To list all possible rational zeros, use the Rational Root Theorem. This theorem states that the possible rational zeros of a polynomial with integer coefficients are given by \( \frac{p}{q} \), where \( p \) is a factor of the constant term (\(-12\) in this case) and \( q \) is a factor of the leading coefficient (\(6\) in this case). The factors of \(-12\) are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).The factors of \(6\) are: \( \pm 1, \pm 2, \pm 3, \pm 6 \).Thus, the possible rational zeros are:\[ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm 3, \pm 4, \pm \frac{2}{3}, \pm \frac{3}{2}, \pm 6, \pm 12 \].

Step 2: Use Graph to Eliminate Possible Zeros

Graph the polynomial function \( P(x) = 6x^3 + 17x^2 - 31x - 12 \) using a graphing calculator or software. Look for points where the graph intersects the x-axis, as these are the possible zeros.Upon inspection of the graph, we can see that the function seems to cross the x-axis around \( x = -3 \), \( x = \frac{1}{2} \), and \( x = 4 \). These can be our initial candidates for the rational zeros.

Step 3: Verify Potential Zeros

Use synthetic division or direct substitution to confirm if the values obtained from the graph are indeed zeros of the polynomial.1. **Testing \( x = -3 \):** Using synthetic division with \(-3\) on \( P(x) \), we find the remainder is \(0\), confirming that \( x = -3 \) is a zero.2. **Testing \( x = \frac{1}{2} \):** Using synthetic division with \(\frac{1}{2}\), the remainder is \(0\), confirming that \( x = \frac{1}{2} \) is also a zero.3. **Testing \( x = 4 \):** Use synthetic division with \(4\), the remainder is \(0\), confirming that \( x = 4 \) is another zero.

Step 4: Factor the Polynomial

Since we have found all rational zeros, we can express the polynomial as follows:\[ P(x) = (x + 3)(x - \frac{1}{2})(x - 4) \]. To have integer coefficients, multiply the entire polynomial by 2, resulting in:\[ P(x) = 6(x + 3)(2x - 1)(x - 4) \]. Perform the multiplication to confirm:First, \((2x - 1)(x - 4) = 2x^2 - 8x - x + 4 = 2x^2 - 9x + 4\)Then, expand:\( 6(x + 3)(2x^2 - 9x + 4)\) results in the original polynomial expression.

Key Concepts

Polynomial Functions

Polynomial functions are a key concept in algebra and are expressed in the form of an equation with one or more terms where the coefficients are real numbers and the exponents are whole numbers. The general form of a polynomial function can be written as \[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0 \]. Each term is made up of a coefficient (e.g., \(a_n, a_{n-1}\)), a variable \(x\), and an exponent. The highest exponent, \(n\), determines the degree of the polynomial, which influences the shape and the number of zeros of the polynomial graph.

• *Constant* term: Coefficient without any variables (e.g., \(a_0\)).
• *Leading Coefficient*: The number in front of the term with the highest exponent (e.g., \(a_n\)).
• *Degree of the Polynomial*: The highest power of the variable in the polynomial.

Understanding polynomial functions is critical when evaluating or transforming expressions such as expressing the polynomial as a product of its factors. They are used to model scenarios in physics, economics, and statistics, where change rates and curve behaviors need to be analyzed. Always remember, the zeros of a polynomial are the solutions to the equation \(P(x) = 0\). These zeros can be determined graphically or through methods like factoring or the Rational Root Theorem.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It simplifies the long division process by focusing only on the coefficients, making it a powerful tool to verify if a number is a zero of the polynomial. When using synthetic division, align the coefficients of the polynomial in descending order of exponents. If a term is missing, a zero is used in its place. Then, proceed with the calculation as follows:1. **Arrange Coefficients:** Write the coefficients in a row. For the polynomial \(6x^3 + 17x^2 - 31x - 12\), you'd write *6, 17, -31, -12*.2. **Set the Divisor:** Place the potential zero outside the row. For example, testing \(x = -3\), place \(-3\) outside.3. **Perform Synthetic Division:** Bring down the first coefficient; then multiply it by the divisor and add the result to the next coefficient. Continue this process till all coefficients are dealt with.

• The remainder tells if the divisor is a zero: 0 means \(x = c\) is a zero.
• Continue with further testing to find all possible zeros.

Synthetic division is not only neat but also efficient, cutting down the process steps and improving accuracy when handling potential rational zeros derived from the Rational Root Theorem.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of its simpler polynomial factors. This is crucial for solving polynomial equations and finding their zeros. Once the zeros are established using methods like graphing or synthetic division, factoring becomes attainable.Given a polynomial function like \(P(x) = 6x^3 + 17x^2 - 31x - 12\), and after identifying its zeros (\(-3, \frac{1}{2}, 4\)) through testing:

• **Express each zero as a factor**: Convert each zero into a factor:
  • -3 becomes \((x + 3)\)
  • \(\frac{1}{2}\) becomes \((2x - 1)\)
  • 4 becomes \((x - 4)\)
• **Combine into a factored form**: Multiply these factors together in a simpler product: \[ P(x) = 6(x + 3)(2x - 1)(x - 4) \]

This step is essential for simplifying and solving polynomial equations as the roots are directly visible. Factoring transforms the equation into a form where setting any factor to zero quickly solves for the root values, clarifying solutions further in calculations or applications.