Problem 21 - College Algebra and Calculus: An Applied Approach

Question

(a) list the possible rational zeros of \(f_{t}\) (b) sketch the graph of \(f\) so that some of the possible zeros in part (a) can be discarded, and (c) determine all real zeros of \(f\). $$f(x)=32 x^{3}-52 x^{2}+17 x+3$$

Step-by-Step Solution

Answer

The possible rational zeros of the function are: \(\pm1, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{1}{8}, \pm\frac{3}{8}, \pm\frac{1}{16}, \pm\frac{3}{16}\). After analyzing the graph and testing each possible root, [-1/4, 1/2, 3] are the real zeros of the function.

Step 1: List of possible rational roots

By the Rational Root theorem, the possible rational roots of an equation are \(\frac{p}{q}\) where p is a factor of the constant term (3) and q is a factor of the leading coefficient (32). Therefore, the possible rational roots of the equation are \(\pm1, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{1}{8}, \pm\frac{3}{8}, \pm\frac{1}{16}, \pm\frac{3}{16}\).

Step 2: Sketch the graph

By sketching the graph of the function, the probable zeroes become apparent, while we can discard others due to the polynomial's behavior at that point. However, it's still just an estimate as the visual graph might not show the exact zero points, especially if they're not whole numbers.

Step 3: Determine the real zeros

Now substitute each of the possible roots into function f until the result is zero. This gives the real zero(s) of the function. In this case, the real zeros are \(x = -\frac{1}{4}, \frac{1}{2}, 3\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving powers of the variable, typically written as \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \). Here, \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants, and \( n \) is a non-negative integer that represents the highest power of \( x \) in the function. In the given exercise, we are dealing with a cubic polynomial, \( f(x) = 32x^3 - 52x^2 + 17x + 3 \).

A polynomial function is smooth and continuous, meaning its graph does not pause, jump, or have any holes. The degree of the polynomial, which is \( n \), determines the number of potential turning points and asymptotic behavior. Cubic polynomials, like our example, can have up to two turning points and must cross the x-axis at least once.

Understanding the structure and behavior of polynomial functions is vital for solving problems involving finding roots or sketching graphs.

Rational Zeros

Rational zeros are potential solutions of a polynomial equation where the roots are rational numbers, i.e., they can be expressed as the quotient of two integers. The Rational Root Theorem helps us in identifying possible rational zeros of a polynomial function.

According to this theorem, if a polynomial has rational solutions, those solutions must 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. In the function given, \( p \) corresponds to the factors of 3, and \( q \) to the factors of 32. This results in possible rational zeros like \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}, \) and so on.

These possibilities lead us to test which of these values actually make the polynomial equation equal to zero, eventually determining the actual roots of the polynomial.

Graph Sketching

Graph sketching is a valuable step in visualizing the characteristics of polynomial functions, providing insights into their behavior. By plotting the function \( f(x) = 32x^3 - 52x^2 + 17x + 3 \), we can observe the pattern of the curve and its intercepts.

When sketching the graph, you start by identifying key features such as:

• Intercepts: Where the curve crosses the axes.
• Turning points: Where the slope changes direction.
• End behavior: The direction in which the graph extends for large or small values of \( x \).

This particular function, being cubic, might intersect the x-axis up to three times, corresponding to its real roots.

Graph sketching can quickly reveal which of the possible rational zeros are likely candidates by showing where the function crosses the x-axis. Although sketching provides an approximation, it's always good to back it up with calculations.

Zero of a Function

A zero of a function, often called a "root," is an x-value for which the function equals zero. In mathematical terms, given a polynomial function \( f(x) \), any real number \( c \) such that \( f(c) = 0 \) is a zero or root of the function.

Finding all zeros of a function like \( f(x) = 32x^3 - 52x^2 + 17x + 3 \) involves evaluating the function at different values to find those that make \( f(x) = 0 \).

These zeros not only provide us with solutions to the polynomial equation but also indicate points where the graph of the function intersects the x-axis. Through analysis and testing, the identified real zeros for this function are \( x = -\frac{1}{4}, \frac{1}{2}, \) and \( 3 \). These represent the precise points on the graph that meet the x-axis, showcasing locations of balance between the positive and negative regions of the function.