Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

A graph of \(y=f(x)\) is shown, where \(f(x)=2 x^{5}-3 x^{4}+x^{3}-8 x^{2}+5 x+3\) and \(f(-x)=-2 x^{5}-3 x^{4}-x^{3}-8 x^{2}-5 x+3\). (a) How many negative real zeros does \(f\) have? Explain. (b) How many positive real zeros are possible for \(f ?\) Explain. What does this tell you about the eventual right-hand behavior of the graph? (c) Is \(x=-\frac{1}{3}\) a possible rational zero of \(f ?\) Explain. (d) Explain how to check whether \(\left(x-\frac{3}{2}\right)\) is a factor of \(f\) and whether \(x=\frac{3}{2}\) is an upper bound for the real zeros of \(f\).

Step-by-Step Solution

Answer

The function \(f\) has 5, 3 or 1 negative real zeros. \(f\) also has 4, 2, or 0 positive real zeros. It's not possible for \(x=-1/3\) to be a rational zero of \(f\). To determine if \((x-\frac{3}{2})\) is a factor of \(f\) and if \(x=\frac{3}{2}\) is an upper bound for the real zeros of \(f\), one should substitute these values into \(f(x)\).

Step 1: Determining number of negative real zeros

The function \(f(-x)\) is a reflection of \(f(x)\) about the y-axis. Comparing \(f(x)\) and \(f(-x)\), it is clear that any positive real zero of \(f(x)\) will become a negative real zero of \(f(-x)\). The function has 5 sign changes from \(f(x)\) to \(f(-x)\), thus it has 5 or 3 or 1 negative real zeros. Therefore, \(f(x)\) has 5, 3, or 1 negative real zeros.

Step 2: Determining number of positive real zeros

The number of positive real zeros is determined by the number of sign changes in \(f(x)\). The function \(f(x)\) shows 4 sign changes, which means there can be 4 or 2 or 0 positive real zeros. Therefore, \(f(x)\) has 4, 2, or 0 positive real zeros. Given the degree of the polynomial is 5, the total number of real zeros is an odd number, so the combination could be 4 positive and 1 negative roots or 2 positive and 3 negative roots or 0 positive and 5 negative roots. This suggests that the right-hand behavior of the graph should be eventually going up and touch the positive part of y-axis.

Step 3: Checking for rational zeros

The Rational Root Theorem states that any rational root, written in lowest terms, has a numerator that is a factor of the constant term, and a denominator that is a factor of the leading coefficient. Here, the constant term is 3 and the leading coefficient is 2. Thus, the possible rational roots are \(\pm\)1, \(\pm\)3, \(\pm\)1/2, and \(\pm\)3/2. \(x=-1/3\) is not among the possible rational roots.

Step 4: Checking for a factor and upper bound

One can check whether \((x-\frac{3}{2})\) is a factor of \(f\) by setting \(x=\frac{3}{2}\) and then substituting it in \(f(x)\). If \(f(\frac{3}{2}) = 0\), then \((x-\frac{3}{2})\) is a factor of \(f\). To check whether \(x=\frac{3}{2}\) is an upper bound for the real zeros of \(f\), we can use the Upper Bound Theorem: If \(a > 0\) and \(f(a) > 0\), then \(a\) is an upper bound for the real zeros of \(f\). Thus, calculate \(f(\frac{3}{2})\) and ascertain if it is a positive value.

Key Concepts

Real Zeros

Real zeros, also known as real roots, are the x-values where a polynomial function equals zero. These are the points where the graph of the function intersects the x-axis. To determine the number of negative or positive real zeros, we can use the function's behavior and other related theorems.
For the function given, the number of negative real zeros is linked with the number of sign changes when comparing the function to its reflected form. With 5 sign changes from \( f(x) \) to \( f(-x) \), we know it can have 5, 3, or 1 negative real zeros. Similarly, the number of positive real zeros is determined by the sign changes in \( f(x) \) itself, which results in possibilities of 4, 2, or 0 positive real zeros.
These possibilities arise from Descartes' Rule of Signs which suggests variations in potential zeros by decreasing in increments of 2.

Rational Root Theorem

The Rational Root Theorem is a useful tool to find possible rational zeros of a polynomial function. This theorem states that if \( p/q \) is a root of a polynomial, then \( p \) must be a factor of the constant term, and \( q \) must be a factor of the leading coefficient.
For the polynomial \( f(x) = 2x^5 - 3x^4 + x^3 - 8x^2 + 5x + 3 \):

• Constant term is 3: Factors are \( \pm 1, \pm 3 \).
• Leading coefficient is 2: Factors are \( \pm 1, \pm 2 \).

This results in potential rational roots of \( \pm 1, \pm 3, \pm 1/2, \pm 3/2 \). The particular case of \( x = -1/3 \), required for evaluation, is not a possibility under this scenario as it isn't part of the list of rational roots.

Sign Changes

Sign changes in a polynomial provide insight into the number of possible real zeros. By observing changes in the signs of the polynomial’s coefficients, we can determine variations in potential real zeros.
From the given polynomial, examining the coefficients:

• The function \( f(x) \) has 4 sign changes.
• The reflected function \( f(-x) \) also has 5 sign changes.

According to Descartes' Rule of Signs:

• \( f(x) \)'s 4 sign changes suggest up to 4 positive real zeros.
• \( f(-x) \)'s 5 subtracts in steps of two, indicating 5, 3, or 1 negative real zeros.

These sign changes help predict the number and types of roots without full calculation of zeros.

Upper Bound Theorem

The Upper Bound Theorem is instrumental in narrowing down potential real zeros of the polynomial. It states that if a number \( a \), larger than zero, leads \( f(a) \) to be greater than zero, then \( a \) serves as an upper bound for polynomial real zeros.
In this context, to check whether \( x = \frac{3}{2} \) is an upper bound, you need to compute \( f(\frac{3}{2}) \):

• Substitute \( x = \frac{3}{2} \) into the polynomial.
• If the result is a positive number, then \( \frac{3}{2} \) is indeed an upper bound.

This approach provides clarity and prevents chasing roots beyond the feasible domain, reflecting an efficient strategy to categorize possible roots of polynomial functions. Assessing whether \( x = \frac{3}{2} \) as a factor further requires checking if it makes the function zero, paired with the concept of upper bounds for a rounded understanding.