Problem 61 - Algebra and Trigonometry

Question

Find the zeros of \(f(x)=(x+3)^{2}(2 x-5)^{3}\) and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the \(x\) -axis and turns around at each zerc-

Step-by-Step Solution

Answer

The zeros of the function \(f(x) = (x+3)^{2}(2x-5)^{3}\) are x = -3 and x = 5/2. -3 has multiplicity 2 and the graph touches the x-axis and turns around at this point. 5/2 has multiplicity 3 and the graph crosses the x-axis at this point.

Step 1: 1. Find the Zeros

To find the zeros of the function \(f(x) = (x+3)^{2}(2x-5)^{3}\), set it equal to zero and solve for \(x\):\n0 = (x+3)^{2}(2x-5)^{3}\nThis can be broken down into two equations due to the factors of the function:\n(x + 3) = 0 and (2x - 5) = 0\nBy solving these equations, we find the roots \(x = -3\) and \(x = 5/2\).

Step 2: 2. Determine the Multiplicities

The multiplicity of a root is determined by the power at which it occurs in the original function.\nThe root -3 occurs to the power of 2 in \(f(x)\) and the root 5/2 occurs to the power of 3. Hence, the root -3 has multiplicity 2 and the root 5/2 has multiplicity 3.

Step 3: 3. Graph Analysis

A function crosses the x-axis at a particular root if the multiplicity of that root is odd and touches the x-axis and turns around if the multiplicity is even.\nSince -3 has multiplicity 2 (even), the graph will touch the x-axis and turn around at \(x = -3\).\nSince 5/2 has multiplicity 3 (odd), the graph will cross the x-axis at \(x = 5/2\).

Key Concepts

Polynomial Factoring

When solving polynomial equations, recognizing patterns in factoring can simplify the process significantly. Factoring is the method of breaking down a polynomial into simpler pieces, known as factors, whose product equals the original polynomial. Think of it as reverse multiplication.

For example, in the polynomial equation given by the function f(x) = (x+3)^2(2x-5)^3, it's evident that the expression is already factored into two parts, (x+3) and (2x-5), each raised to a certain power. The key in factoring is to identify the base of each factor that gives you zero when plugged into the equation. To solve for zeros, or roots, we set each factor equal to zero and solve for x. Since multiplication is at the heart of factors, if one factor equals zero, the entire product will be zero, thus revealing the zeros of the original polynomial.

Multiplicity of Zeros

The concept of multiplicity relates to how many times a particular zero occurs in a polynomial function. A zero's multiplicity affects the polynomial's behavior at the x-coordinate intersecting that zero.

In our exercise, the polynomial f(x) = (x+3)^2(2x-5)^3 presents two zeros with differing multiplicities. The zero at x = -3 results from the factor (x+3) and occurs with a multiplicity of 2 because the factor is raised to the second power. The zero at x = 5/2 has a multiplicity of 3 due to the factor (2x-5) being raised to the third power. A general rule is that if a zero has an even multiplicity, the graph of the polynomial merely touches the x-axis at that zero and turns around; it does not cross the axis. On the other hand, a zero with an odd multiplicity will mean the polynomial crosses the x-axis, creating a more distinctive visual change in direction.

Graph Analysis

Understanding the zeros and their multiplicities gives insights into the graph of the polynomial. However, graph analysis goes beyond just plotting points; it's about recognizing how the polynomial behaves between and at the zeros.

In the function f(x) = (x+3)^2(2x-5)^3, the zero at x = -3, with an even multiplicity, indicates that the graph of f(x) will approach the x-axis, gently touch it (imagine a soft bounce), and then head back in the direction it came from. This point is often referred to as a point of tangency. Conversely, at x = 5/2, where the zero has an odd multiplicity of 3, the graph will intersect the x-axis. As it passes through, there can be a noticeable change in curvature, since the function 'crosses' the axis and continues on the opposite side. Graph analysis in this context is a powerful tool to visually represent the abstract algebraic concepts inherent in polynomial functions.