Answered

Find the root(s) of $f(x)=(x+5)^3(x-9)^2(x+1)$. Choose three correct answers.


  • 9 with multiplicity 2
  • -1 with multiplicity 1
  • 5 with multiplicity 3
  • -1 with multiplicity 0
  • -5 with multiplicity 3
  • 1 with multiplicity 0
  • 1 with multiplicity 1
  • -9 with multiplicity 2

Verified Answer :

  • Analyze the polynomial \(f(x) = (x+5)^3(x-9)^2(x+1)\) by examining each factor.
  • Set each factor to zero to find the roots: \(x = -5, 9, -1\).
  • Determine the multiplicity from the exponent of each factor: 3, 2, 1 respectively.
  • Match these results with the provided options: \(\boxed{9 \text{ with multiplicity } 2, -1 \text{ with multiplicity } 1, -5 \text{ with multiplicity } 3}\).

Explanation

  1. Analyzing the Polynomial Structure

To find the roots of the polynomial \(f(x) = (x+5)^3(x-9)^2(x+1)\), we look for the values of \(x\) that make the function equal to zero. A polynomial in factored form \(f(x) = c(x-r_1)^{m_1}(x-r_2)^{m_2}\dots(x-r_k)^{m_k}\) has roots at \(r_1, r_2, \dots, r_k\), where \(m_i\) represents the multiplicity of each root. The multiplicity indicates how many times the corresponding factor appears in the expression.

  1. Identifying Roots and Multiplicities

We identify the individual factors of the polynomial and determine the root and its multiplicity for each:

  • For the factor \((x+5)^3\), we set \(x+5 = 0\), which gives \(x = -5\). Since the exponent is 3, the root \(-5\) has a multiplicity of 3.
  • For the factor \((x-9)^2\), we set \(x-9 = 0\), which gives \(x = 9\). Since the exponent is 2, the root \(9\) has a multiplicity of 2.
  • For the factor \((x+1)\), which is the same as \((x+1)^1\), we set \(x+1 = 0\), which gives \(x = -1\). Since the exponent is 1, the root \(-1\) has a multiplicity of 1.
  1. Selecting the Correct Options

Based on our analysis, the roots of the polynomial \(f(x) = (x+5)^3(x-9)^2(x+1)\) are:

  • \(x = 9\) with multiplicity \(2\)
  • \(x = -1\) with multiplicity \(1\)
  • \(x = -5\) with multiplicity \(3\)

Comparing these to the given options, we select:

  • 9 with multiplicity 2
  • -1 with multiplicity 1
  • -5 with multiplicity 3

Examples

In physics, roots of polynomials are often used to determine equilibrium points where a force or potential energy might be zero. For instance, in a system where the position of an object is modeled by a polynomial, finding the roots allows us to identify the moments in time or positions where the net force is zero. Understanding multiplicities also tells us the behavior of the system near those points—whether it passes through zero or touches and bounces back, which is crucial for analyzing stability in mechanical or electrical engineering.

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