Problem 78 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6 $$

Step-by-Step Solution

Answer

Rational zeros: \( x = \frac{3}{2}, \frac{1}{2}, -1 \); irrational zeros: \( x = \frac{5 \pm i}{4} \).

Step 1: Apply the Rational Zeros Theorem

The Rational Zeros Theorem states that any rational zero of the polynomial \( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \) is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \). Here, \( a_0 = 6 \) and \( a_n = 8 \), so the possible rational zeros are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4} \).

Step 2: Evaluate Potential Rational Zeros

Substitute possible rational zeros into \( P(x) \) to see if they make \( P(x) = 0 \). Upon testing, the rational zeros are found to be \( x = \frac{3}{2} \), \( x = \frac{1}{2} \), and \( x = -1 \).

Step 3: Perform Polynomial Division

Using synthetic division, divide \( P(x) \) by \( x - \frac{3}{2} \), \( x - \frac{1}{2} \), and \( x + 1 \), the known rational zeros, to simplify \( P(x) \). After division, the resulting polynomial is \( 8x^2 - 10x + 4 \).

Step 4: Find Remaining Zeros Using Quadratic Formula

Solve the quadratic \( 8x^2 - 10x + 4 = 0 \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 8 \), \( b = -10 \), and \( c = 4 \). Calculating, we find the roots are \( x = \frac{5 \pm i}{4} \), which are irrational.

Key Concepts

Polynomials and Polynomial Division

A polynomial is an expression consisting of variables, coefficients, and exponents that are combined using addition, subtraction, multiplication, and non-negative integer exponents. Polynomial division is a technique used to divide one polynomial by another. It works in a manner similar to long division with numbers. During the process, you identify a term that, when multiplied by the divisor, yields the highest degree term of the dividend. Subtract to find the remainder, and continue dividing. This is repeated until the remainder is a lower degree than the divisor, or until it is zero.

This method helps in simplifying polynomials, especially useful when initially solving for zeros like in our problem. By performing polynomial division using known zeros, we can reduce a higher degree polynomial down to a quadratic, making it simpler to solve the remaining roots.

Synthetic Division Simplified

Synthetic division is a streamlined method of dividing a polynomial by a binomial of the form \(x - c\). It is particularly useful when evaluating polynomials at specific values or when simplifying them before using other techniques, like the quadratic formula.

To perform synthetic division:

• Write down the coefficients of the polynomial in descending order of degree.
• Use the zero you're testing (from your list of possible rational zeros) as the 'divisor' and write it on the left of the row of coefficients.
• Bring down the leading coefficient to start the process.
• Multiply the divisor by this leading coefficient, placing the result under the next coefficient, then add the two together.

Continue this process across all coefficients. If the last number (remainder) is zero, then the divisor is indeed a zero of the polynomial.

Synthetic division is less cumbersome and faster than traditional polynomial division, providing quick results especially in identifying zeros.

Exploring with the Quadratic Formula

The quadratic formula is a universal tool used to find the zeros of a quadratic equation \(ax^2 + bx + c = 0\). The formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

This tool is particularly useful when other factoring methods fail or when the quadratic is not easily factorable.

For the quadratic \(8x^2 - 10x + 4 = 0\) obtained in the problem, the values can be substituted into the formula:

• \(a = 8\)
• \(b = -10\)
• \(c = 4\)

Calculating yields the solutions \(x = \frac{5 \pm i}{4}\), indicating the presence of complex and irrational zeros, revealing more about the nature of the polynomial's roots.

Using Descartes’ Rule of Signs

Descartes’ Rule of Signs is a powerful technique to predict the number of positive and negative real roots in a polynomial equation. By counting the number of sign changes in the sequence of the polynomial's coefficients, you can determine the maximum number of positive real roots.

To use Descartes’ Rule of Signs:

• List the coefficients of the polynomial from highest to lowest degree.
• Check the sequence and count how often the signs change from positive to negative or vice versa.

For negative roots, replace \(x\) with \(-x\) in the polynomial and repeat the sign change count.

This simple rule can help narrow down the possible number of real zeros you might find, guiding your choice of techniques such as the Rational Zeros Theorem or synthetic division in solving polynomials.