Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=x^{4}-x^{3}-29 x^{2}-x-30$$

Step-by-Step Solution

Answer

After using the Descartes' Rule of Sign, Rational Root Theorem, and graphing, there can be inferred that all the real zeros of the given function \(f(x)=x^{4}-x^{3}-29 x^{2}-x-30\) are -1, 1, -6, and 5.

Step 1: Apply Descartes' Rule of Signs

Use Descartes's Rule of Signs to determine the number of positive and negative zeros in the equation. First, identify the number of sign changes in the equation to determine the number of possible positive real zeros. Secondly, replace \(x\) by \(-x\) and identify the number of sign changes in the new equation to determine the number of possible negative real zeros.

Step 2: List Possible Rational Zeros

Determine the possible rational zeros using the Rational Root Theorem, which says the possible rational zeros of a polynomial are of the form \( \pm \frac{p}{q} \), where \(p\) divides the constant term and \(q\) divides the leading coefficient.

Step 3: Graph the Function

Use a graphing utility to graph the function \(f(x)=x^{4}-x^{3}-29 x^{2}-x-30\). By using the graph, certain possibilities for zeros identified in previous steps can be disregarded.

Step 4: Determine all Real Zeros

Examine the graph to locate the real zeros. Then, use synthetic division or the Rational Root Theorem to confirm these zeros.

Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful technique to determine how many positive and negative real zeros a polynomial function might have. To find the potential positive real zeros, you must look at the polynomial and count the number of times the signs of the coefficients change as you move from one term to the next. Each sign change correlates to a possible positive real zero.For instance, in the polynomial function \(f(x) = x^4 - x^3 - 29x^2 - x - 30\), begin by observing the sign changes:- From \(+1\) for \(x^4\) to \(-1\) for \(x^3\) (1 change)- From \(-1\) for \(x^3\) to \(-29\) for \(x^2\) (no change)- From \(-29\) for \(x^2\) to \(-1\) for \(x\) (no change)- From \(-1\) for \(x\) to \(-30\) (no change)This example shows one sign change, suggesting up to one positive real zero.To find possible negative real zeros, replace \(x\) with \(-x\) in the function and analyze sign changes in the transformed polynomial. The number of sign changes in the new polynomial indicates the potential number of negative real zeros.

Rational Root Theorem

The Rational Root Theorem is a handy tool to list all possible rational zeros of a polynomial. It tells us that if a polynomial has any rational zeros, they must be of the form \( \pm \frac{p}{q} \), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.With the polynomial \(f(x)=x^4-x^3-29x^2-x-30\), the factors of the constant term (-30) are:

• \(\pm 1\)
• \(\pm 2\)
• \(\pm 3\)
• \(\pm 5\)
• \(\pm 6\)
• \(\pm 10\)
• \(\pm 15\)
• \(\pm 30\)

The leading coefficient is \(1\), whose only factors are \(\pm 1\). Therefore, the possible rational zeros are exactly the values \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 5\), \(\pm 6\), \(\pm 10\), \(\pm 15\), and \(\pm 30\).Using this method, we have a structured way of identifying potential rational solutions to test on the polynomial.

Synthetic Division

Synthetic division is an efficient method for dividing a polynomial by a linear factor of the form \((x-c)\) where \(c\) is a constant, and is particularly useful when verifying potential zeros of polynomials found using the Rational Root Theorem.To apply synthetic division, follow these steps:

• Write down the coefficient of each term of the polynomial.
• Place the potential zero (\(c\)) to the left and perform operations as follows:
• Bring down the leading coefficient.
• Multiply \(c\) by the result, and add it to the next coefficient.
• Repeat the process through all coefficients.
• If the final result (remainder) is zero, \(c\) is a root of the polynomial.

For example, if you test \(c=3\) on \(f(x)=x^4-x^3-29x^2-x-30\), and synthetic division yields a remainder of zero, then \(x=3\) is a root of \(f(x)\). This method helps confirm rational zeros effectively and also simplifies polynomial degree when trying to factor further.