Problem 49 - 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 $$ 4 x^{4}-x^{3}+5 x^{2}-2 x-6-0 $$

Step-by-Step Solution

Answer

The roots of the polynomial \(4 x^{4}-x^{3}+5 x^{2}-2 x-6=0\) are \(x=-1, x=1/2\), and two roots derived from solving the obtained quadratic polynomial.

Step 1: Identify Potential Rational Roots

According to the Rational Zero theorem, the possible rational zeros of a polynomial function \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_{1}x + a_0\) is \(\pm\) factor of \(a_0 / a_n\). Here, \(4 x^{4}-x^{3}+5 x^{2}-2 x-6\) has a leading coefficient \(a_n\) of 4 and a constant term \(a_0\) of -6, so the possible rational roots are \(\pm\) factors of 6/4, namely \(\pm1, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm2, \pm3\).

Step 2: Use Descartes's Rule of Signs

We determine the number of sign changes in the polynomial function, which can give us the maximum number of positive roots. For the given function \(4 x^{4}-x^{3}+5 x^{2}-2 x-6\), there are 3 sign changes. Hence, there could be 3 or (3-2*1) = 1 positive root. Next, we substitute \(x=-x\) into the polynomial to get the function for negative x. Counting the sign changes in this new polynomial will give us an idea of the possible number of negative roots. The new function becomes \(4 x^{4}-x^{3}-5 x^{2}+2 x-6\) and this has one sign change, so there is 1 or (1-2*0) = 1 negative root.

Step 3: Finding the Actual Roots

We substitute the potential rational roots into the function until we find the roots that make the function equal to zero. Doing so, we find the roots \(x=-1, x=1/2\).

Step 4: Use polynomial division to find the remaining roots

After finding the first two roots, we divide the original polynomial by the corresponding factors \((x-(-1))\) and \((x-(1/2))\) to get the quadratic polynomial. We then solve the quadratic polynomial using the quadratic formula to find the remaining roots.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a powerful tool for finding potential zeros of a polynomial function. It states that if a polynomial has any rational roots (zeros), those roots will be of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
This theorem helps us narrow down the list of possible rational roots, turning a potentially endless search into a manageable list of candidates.

• For a polynomial like \(4x^4 - x^3 + 5x^2 - 2x - 6\), with a constant term \(-6\) and a leading coefficient of \(4\), the candidate roots based on factors would be \(\pm1, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm2, \pm3\).

This simplification is crucial because it allows us not to test every possible number but just a finite set of candidates.

Descartes's Rule of Signs

Descartes's Rule of Signs is a method to predict the number of positive and negative real roots in a polynomial by examining the number of sign changes in its coefficients.
For positive roots, count how many times the sign changes in the polynomial's standard form.
For example, in the polynomial \(4x^4 - x^3 + 5x^2 - 2x - 6\), there are three sign changes:

• From \(+4\) to \(-1\)
• From \(-1\) to \(+5\)
• From \(+5\) to \(-2\)

Thus, there could be 3 or 1 positive root(s) (by Descartes's Rule, the count may decrease by multiples of two).
To determine potential negative roots, substitute \(x\) with \(-x\) in the polynomial, resulting in \(4x^4 - x^3 - 5x^2 + 2x - 6\). There is one sign change here, suggesting 1 negative root.
This rule doesn't tell us which numbers are roots but gives insight into how many might be positive or negative.

Polynomial Division

Once potential roots have been identified using the Rational Zero Theorem and confirmed by substitution into the polynomial, Polynomial Division is used to simplify the polynomial.
For instance, after confirming \(x = -1\) and \(x = \frac{1}{2}\) as roots, divide the original polynomial by \((x + 1)\) and \((2x - 1)\) to find the quotient polynomial.
Polynomial division is like long division for numbers, and it helps reduce the polynomial to a lower degree.

• This makes finding any remaining roots much more straightforward because once simplified, you're left with a polynomial that's easier to manage, often reducible to a quadratic polynomial suitable for other root-finding methods.

It’s a vital step to break down a complex polynomial problem into smaller, manageable parts.

Quadratic Formula

The Quadratic Formula is a time-tested method used to find the zeros of any quadratic equation \(ax^2 + bx + c = 0\). The formula is given as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
After using polynomial division, if you find a quadratic polynomial, this formula becomes your go-to tool.
It takes the coefficients from the simplified quadratic and provides the roots directly, including complex ones.

• For example, after dividing the original polynomial and obtaining a quadratic component, substitute \(a\), \(b\), and \(c\) from it into the quadratic formula for a straightforward solution.

This formula is favored for its reliability and efficiency, especially in cases where factoring becomes challenging or when you're left with irrational or complex solutions.