Answered

Choose all the binomials that are factors of the polynomial function.

$f(x)=x^3-2 x^2-43 x+140$


  • $x+7$
  • $x+4$
  • $x-7$
  • $x-4$
  • $x+5$
  • $x-5$

Verified Answer :

  • Use the Factor Theorem: a binomial \((x-c)\) is a factor of \(f(x)\) if \(f(c)=0\).
  • Test each candidate binomial:
  • \(f(-7) = 0 \implies (x+7)\) is a factor.
  • \(f(4) = 0 \implies (x-4)\) is a factor.
  • \(f(5) = 0 \implies (x-5)\) is a factor.
  • Verify the results confirm \(f(x) = (x+7)(x-4)(x-5)\).
  • Final factors: \(\boxed{x+7, x-4, x-5}\).

Explanation

  1. Analyzing the Problem and Methodology

The objective is to determine which of the given binomials—\((x+7), (x+4), (x-7), (x-4), (x+5), (x-5)\)—are factors of the cubic polynomial function \(f(x) = x^3 - 2x^2 - 43x + 140\). According to the Factor Theorem, a binomial of the form \((x-c)\) is a factor of \(f(x)\) if and only if \(f(c) = 0\). For a binomial of the form \((x+c)\), we evaluate the polynomial at \(-c\) to test if \(f(-c) = 0\). Since the polynomial is of degree 3, we expect to find at most three linear factors.

  1. Testing Candidate Binomials

We systematically apply the Factor Theorem to each candidate binomial:

  1. For \((x+7)\), test \(f(-7)\):

\(f(-7) = (-7)^3 - 2(-7)^2 - 43(-7) + 140 = -343 - 2(49) + 301 + 140 = -343 - 98 + 441 = -441 + 441 = 0\).
Since \(f(-7) = 0\), \((x+7)\) is a factor.

  1. For \((x+4)\), test \(f(-4)\):

\(f(-4) = (-4)^3 - 2(-4)^2 - 43(-4) + 140 = -64 - 2(16) + 172 + 140 = -64 - 32 + 312 = -96 + 312 = 216 eq 0\).
Since \(f(-4) eq 0\), \((x+4)\) is not a factor.

  1. For \((x-7)\), test \(f(7)\):

\(f(7) = (7)^3 - 2(7)^2 - 43(7) + 140 = 343 - 2(49) - 301 + 140 = 343 - 98 - 301 + 140 = 84 eq 0\).
Since \(f(7) eq 0\), \((x-7)\) is not a factor.

  1. For \((x-4)\), test \(f(4)\):

\(f(4) = (4)^3 - 2(4)^2 - 43(4) + 140 = 64 - 2(16) - 172 + 140 = 64 - 32 - 172 + 140 = -140 + 140 = 0\).
Since \(f(4) = 0\), \((x-4)\) is a factor.

  1. For \((x+5)\), test \(f(-5)\):

\(f(-5) = (-5)^3 - 2(-5)^2 - 43(-5) + 140 = -125 - 2(25) + 215 + 140 = -125 - 50 + 355 = -175 + 355 = 180 eq 0\).
Since \(f(-5) eq 0\), \((x+5)\) is not a factor.

  1. For \((x-5)\), test \(f(5)\):

\(f(5) = (5)^3 - 2(5)^2 - 43(5) + 140 = 125 - 2(25) - 215 + 140 = 125 - 50 - 215 + 140 = -140 + 140 = 0\).
Since \(f(5) = 0\), \((x-5)\) is a factor.

  1. Conclusion

Based on the evaluation of each binomial using the Factor Theorem, we found that \(f(-7) = 0\), \(f(4) = 0\), and \(f(5) = 0\). Thus, the binomials that are factors of the polynomial \(f(x) = x^3 - 2x^2 - 43x + 140\) are \((x+7)\), \((x-4)\), and \((x-5)\). The product of these factors confirms our result: \((x+7)(x-4)(x-5) = (x+7)(x^2 - 9x + 20) = x^3 - 9x^2 + 20x + 7x^2 - 63x + 140 = x^3 - 2x^2 - 43x + 140\). The correct binomials are \((x+7)\), \((x-4)\), and \((x-5)\).

Examples

Understanding how to identify factors of a polynomial is essential for simplifying algebraic expressions, solving equations, and analyzing geometric properties like finding the intercepts of a curve. For example, in engineering, when analyzing the stability of a system, one often needs to determine the "roots" of a characteristic equation to understand how the system behaves over time. Knowing which binomials are factors allows us to decompose complex systems into simpler, manageable parts, making it easier to design reliable structures and circuits.

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