Problem 42 - 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. $$ 4 x^{5}+12 x^{4}-41 x^{3}-99 x^{2}+10 x+24=0 $$

Step-by-Step Solution

Answer

The zeros of the polynomial are -1, -2, 2, 3/2, and 1/2.

Step 1: Apply Rational Zero Theorem

Start with the Rational Zero Theorem to list out possible rational zeros. The Rational Zero Theorem says any rational zero of a polynomial will be a fraction in which the numerator is a factor of the constant term (24 in this case) and the denominator is a factor of the leading coefficient (4 in this case). List possible factors of 24 and 4.

Step 2: List out possible rational zeros

By listing the possible factors of 24 (±1, 2, 3, 4, 6, 8, 12, 24) and of 4 (±1, 2, 4), we can determine that the possible rational zeros are ±1, ±1/2, ±2, ±3, ±3/2, ±4, ±6, ±8, ±12, ±24

Step 3: Apply Descartes's Rule of Signs

Use Descartes's Rule of Signs to predict the number of positive and negative real zeros. This involves counting the number of sign changes in the polynomial. Counting sign changes in given polynomial, there are 5 sign changes, hence up to 5 positive real zeros. If you substitute \(x\) by \(-x\), the equation becomes \(4x^{5}-12x^{4}-41x^{3}+99x^{2}+10x-24\=0\) with 2 sign changes, thus up to 2 negative real zeros.

Step 4: Find the zeros

At this stage, you could plug each potential zero from Step 2 into the polynomial to see if it equals zero; however, a quicker way is to use a graphing tool to visualize and find the zeros. The graphing method helps spot that \(x = -1, -2, 2, 3/2\), and \(1/2\) are zeros of the polynomial.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a helpful tool when dealing with polynomial functions. It allows us to list potential rational zeros of a polynomial equation. To use it, identify two key parts of your polynomial: the constant term (the term with no variable) and the leading coefficient (the coefficient of the term with the highest power).

• The numerator of potential rational zeros must be a factor of the constant term.
• The denominator must be a factor of the leading coefficient.

For a polynomial like \(4x^5 + 12x^4 - 41x^3 - 99x^2 + 10x + 24 = 0\), the constant term is 24 and the leading coefficient is 4. So, we list the factors of 24 (\(\pm 1, 2, 3, 4, 6, 8, 12, 24\)) and the factors of 4 (\(\pm 1, 2, 4\)). Cross-matching these gives us a list of possible rational zeros: \(\pm 1, \pm \frac{1}{2}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\). From this list, we can begin checking these candidates to find actual zeros of the polynomial.

Descartes's Rule of Signs

Descartes's Rule of Signs gives us a way to predict how many positive and negative real zeros a polynomial might have. Here’s how it works:

• Count the number of sign changes in the polynomial terms when they are in descending order.
• This count represents the maximum number of positive real zeros possible. It could also be fewer than this number by an even integer.

For negative zeros, substitute \(-x\) for \(x\) and again count the sign changes.
With our given polynomial \(4x^5 + 12x^4 - 41x^3 - 99x^2 + 10x + 24\), observe the signs:

• From \(4x^5\) to \(12x^4\) (no change), \(12x^4\) to \(-41x^3\) (change), and so on.
• We find 5 sign changes, suggesting up to 5 positive real zeros.

For \(-x\), substitute and you find 2 sign changes, indicating up to 2 negative real zeros. This rule helps narrow down our search for real zeros.

Graphing Polynomial Functions

Using a graphing tool to analyze polynomial functions is both practical and insightful. Graphs provide a visual representation, so we can spot where the function crosses the x-axis—these points are the real zeros of the polynomial.
In our example, the polynomial \(4x^5 + 12x^4 - 41x^3 - 99x^2 + 10x + 24\), we utilized a graphing utility to plot the function. Once graphed, the function revealed particular points where it intersected the x-axis:

• \(x = -1\)
• \(x = -2\)
• \(x = 2\)
• \(x = \frac{3}{2}\)
• \(x = \frac{1}{2}\)

These intersections confirm the zeros found using algebraic methods, ensuring a comprehensive understanding of the polynomial's behaviour. Graphing complements manual calculations and provides a strong confirmation of theoretical outcomes.