Problem 32 - College Algebra

Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 3 x^{3}-8 x^{2}-8 x+8=0 $$

Step-by-Step Solution

Answer

The zeros of the polynomial \(3x^{3}-8x^{2}-8x+8=0\) are \(x=-1\), \(x=\frac{4}{3}\), and \(x=-2\).

Step 1: Potential Rational Roots

The Rational Zero Theorem provides potential rational roots as the divisors of the constant (-8) divided by the divisors of the leading coefficient (3). The potential rational roots are \( \pm\frac{1}{3} \), \( \pm\frac{2}{3} \), \( \pm\frac{4}{3} \), \( \pm\frac{8}{3} \), \( \pm1 \), \( \pm2 \), \( \pm4 \), and \( \pm8 \).

Step 2: Number of Positive or Negative Real Roots

Descartes's Rule of Signs involves counting the change in signs in the polynomial to determine the maximum number of positive or negative real roots. From \(3x^{3}-8x^{2}-8x+8\), there's one change in sign, which means there's exactly one positive real root.

Step 3: Synthetic Division

By trial and error and using the Rational Zeros Theorem, it is found that a root of the polynomial is \(x=-1\). Now use synthetic division to find the other roots. Divide \(3x^{3}-8x^{2}-8x+8\) by \(x+1\) to reduce the degree of the polynomial. This yields \(3x^{2}-11x-8\).

Step 4: Finding the Remaining Roots

After synthetic division, the resulting polynomial is \(3x^{2}-11x-8\), which is a quadratic polynomial that can be solved using the Quadratic Formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Doing this gives the two remaining roots \(x=\frac{4}{3}\) and \(x=-2\)

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a valuable tool used to determine potential rational solutions of a polynomial equation. It works by examining the constant term and the leading coefficient of the polynomial.

Here's how it functions:

• The roots of the constant term (numerator) are potential numerators of the rational roots.
• The roots of the leading coefficient (denominator) are potential denominators of the rational roots.

For example, in the polynomial equation given as \[3x^3 - 8x^2 - 8x + 8 = 0,\] the constant term is 8 and the leading coefficient is 3.
Plausible rational roots are created by pairing each factor of the constant (1, 2, 4, 8) with each factor of the leading coefficient (1, 3), resulting in values like \( \pm\frac{1}{3}, \pm1, \pm2, \pm4, \) and so forth.

Descartes's Rule of Signs

Descartes's Rule of Signs helps predict the number of positive and negative real roots in a polynomial. It involves counting the number of sign changes in the sequence of coefficients.
This straightforward rule states:

• The number of positive real roots is the same as the number of sign changes in the polynomial, or less by an even number.
• The number of negative real roots is determined by the sign changes in the sequence of coefficients when \(x\) is replaced by \(-x\).

As an illustration, for the polynomial \(3x^3 - 8x^2 - 8x + 8\), analyzing the sign changes shows there's exactly one positive real root since there is one sign change.

Synthetic Division

Synthetic Division is an efficient method for dividing a polynomial by a binomial of the form \( x - c \). It drastically simplifies the division process compared to long division.
Synthetic division saves time and reduces the potential for calculation errors. Here’s a brief overview of how it works:

• The value of \(c\) is used as the divisor.
• Only the coefficients of the polynomial are used in the computation.
• This method involves listing coefficients, performing arithmetic operations, and ultimately obtaining the quotient and remainder.

In the provided example, dividing \(3x^3 - 8x^2 - 8x + 8\) by \((x+1)\) found a quotient of \(3x^2 - 11x - 8\), reducing the problem to a simpler quadratic form to find other roots.

Quadratic Formula

The Quadratic Formula is a well-known formula applied to find the roots of quadratic equations, which are polynomials of degree 2.
The formula used is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
In our scenario, after employing synthetic division, we have the quadratic equation \(3x^2 - 11x - 8 = 0\).
Utilizing the quadratic formula calculates the roots, \(x = \frac{4}{3}\) and \(x = -2\), thoroughly solving for all zeros of the initial polynomial.