Problem 52 - College Algebra

Question

In Exercises \(39-52\), 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 $$ 4 x^{5}+12 x^{4}-41 x^{3}-99 x^{2}+10 x+24-0 $$

Step-by-Step Solution

Answer

[1, 2, -3, \(1+ \sqrt{2}\), \(1- \sqrt{2}\)] are the zeros of the polynomial.

Step 1: Potential Rational Roots using the Rational Zero Theorem

The Rational Zero Theorem lets us find all possible rational zeros of the polynomial, which are of the form \(\pm p/q\) where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. Here, the constant term is 24 and the leading coefficient is 4. The possible rational zeros are \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm24\).

Step 2: Descartes' Rule of Signs

Descartes' Rule of Signs is used to determine the possible number of positive and negative real roots. The rule count the number of sign changes in the polynomial's terms. We get 3 sign changes, so the number of positive real roots could be 3, or it could be less by a positive even number, so 1. To count negative roots, apply the rule to the polynomial with \(x\) replaced by \(-x\). We get 2 sign changes, meaning there could be 2 or 0 negative roots.

Step 3: Try out Potential Rational Roots

Use synthetic division or direct evaluation of the polynomial at the potential rational roots. It's quickly found that 1, 2, -3 are zeros of the polynomial. After factoring them out, we get a quadratic polynomial, which can also be solved for zeros.

Step 4: Solve the Quadratic Polynomial

Solve the reduced polynomial \(4x^2 - 8x -8 = 0\) . Use quadratic formula, which states the roots are given by \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The roots of this quadratic are \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\).

Step 5: Check the answer

Substitute all obtained solutions into the original equation to verify. All solutions will make the equation equal to zero, confirming the result.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a helpful tool, particularly with polynomial equations. It helps us to identify all potential rational zeros. These zeros are in the form \( \pm \frac{p}{q} \), where \( p \) is a factor of the polynomial's constant term and \( q \) is a factor of the leading coefficient.
For the polynomial \( 4x^5 + 12x^4 - 41x^3 - 99x^2 + 10x + 24 = 0 \), we have a constant term of 24, and a leading coefficient of 4.
This makes the list of potential zeros:

• \( \pm 1 \)
• \( \pm 2 \)
• \( \pm 3 \)
• \( \pm 4 \)
• \( \pm 6 \)
• \( \pm 8 \)
• \( \pm 12 \)
• \( \pm 24 \)

The Rational Zero Theorem provides us with a starting point, but we still need to verify which, if any, of these potential values are actual zeros of the polynomial.
By evaluating or using other methods like synthetic division, we can confirm the true zeros.

Descartes' Rule of Signs

Descartes' Rule of Signs allows us to predict the number of positive and negative real roots a polynomial equation might have. These predictions are based on sign changes in the polynomial's coefficients.
First, for positive roots, you count the sign changes in the polynomial as it is.

• In our equation, the sequence is: \( +, +, -, -, +, + \) which results in 3 sign changes.
• This suggests the possibilities for positive roots could be 3 or reduced by a positive even number, so **1**.

For negative roots, substitute \( x \) with \( -x \) and calculate the sign changes:

• When we do this, the sequence shows 2 sign changes.
• This means there could be 2 or 0 negative real roots.

Descartes’ Rule of Signs provides valuable insight into the nature of the roots we might expect, helping narrow down the possibilities before diving into labor-intensive calculations.

Synthetic Division

Synthetic Division is a simplified method to divide a polynomial by a binomial of the form \( x - r \). It's particularly useful in verifying potential zeros of a polynomial, as suspected from the Rational Zero Theorem.
The process allows for a streamlined form of division that is less bulky than long division.
Let's say we're testing whether a potential root \( r = 2 \) is a zero of the polynomial.

• We write down the coefficients: 4, 12, -41, -99, 10, 24.
• By doing synthetic division with 2, we find the remainder.
• If the remainder is zero, \( r = 2 \) is indeed a root.

Through this method, when root candidates 1, 2, and -3 were tested, they divided the polynomial perfectly, confirming their validity as zeros.
Once zeros are identified, they can be factored out, ultimately reducing the degree of the polynomial and leading to the remaining quadratic part.

Quadratic Formula

After reducing the initial polynomial using Synthetic Division and confirming certain roots, you might be left with a simpler quadratic equation.
This is where the Quadratic Formula comes in handy.
For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the formula to find its roots is:\[\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the original exercise, after factoring out known zeros, we have the quadratic \( 4x^2 - 8x - 8 = 0 \).
Using the quadratic formula:

• \( a = 4 \), \( b = -8 \), and \( c = -8 \).
• The discriminant \( b^2 - 4ac = (-8)^2 - 4(4)(-8) = 32 \).
• Solving yields roots: \( 1 + \sqrt{2} \) and \( 1 - \sqrt{2} \).

The Quadratic Formula is a robust method ensuring you find all real roots of a quadratic equation, especially when factoring isn't obvious.