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Chapter 3: Problem 32

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

Short Answer

Expert verified
The zeros of the polynomial function \(f(x) = x^3 + 5x^2 - 9x - 45\) are \(x = -5, 9, -1\), each with a multiplicity of 1. The graph crosses the x-axis at each of the three points.

Step by step solution

01

Identify the Polynomial

The given polynomial is \(f(x) = x^3 + 5x^2 - 9x - 45\).
02

Use Polynomial Division or Factoring to Find Zeros

The polynomial \(f(x)\) can be factored as follows \(f(x) = (x+5)(x-9)(x+1)\). This recognizes that when \(f(x) = 0\): \(x+5 = 0\), \(x-9 = 0\), or \(x+1 = 0\). Solving these equations for \(x\), we find that \(x = -5, 9, -1\). Thus, the zeros of the polynomial are \(x = -5, 9, -1\).
03

Determine the Multiplicity of Each Zero

The zeros \(x = -5, 9, -1\) all appear only once in the factored function. This means each zero has a multiplicity of 1.
04

Determine if the Graph Crosses the X-Axis or Touches the X-Axis

Since the multiplicity of all zeros is odd, the graph crosses the x-axis at all three points: \(x = -5, 9, -1\).

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