Verified Answer :

  • A polynomial is completely factored if it is expressed as a product of irreducible polynomials, meaning no further non-trivial factors can be extracted.
  • We analyzed each option:
  • \(g^5 - g\) factors to \(g(g - 1)(g + 1)(g^2 + 1)\).
  • \(4g^3 + 18g^2 + 20g\) factors to \(2g(2g + 5)(g + 2)\).
  • \(24g^2 - 6g^4\) factors to \(6g^2(2 - g)(2 + g)\).
  • We determined that \(2g^2 + 5g + 4\) is irreducible because its discriminant \(\Delta = -7\) is negative, meaning it has no real roots and cannot be factored further.
  • The expression already factored completely is \(\boxed{2g^2 + 5g + 4}\).

Explanation

  1. Definition of Factored Completely

A polynomial is considered "factored completely" when it is expressed as a product of irreducible polynomials over the field of rational numbers (or integers). This means that no more factors can be extracted, and each factor in the product cannot be broken down any further into simpler polynomial components. We will evaluate each given expression to determine if it can be factored further.

  1. Analyzing Each Polynomial Option
  2. For \(g^5 - g\):

First, we extract the greatest common factor, which is \(g\):
\[g(g^4 - 1)\]
The expression \(g^4 - 1\) is a difference of squares: \((g^2)^2 - 1^2 = (g^2 - 1)(g^2 + 1)\).
Furthermore, \(g^2 - 1\) is another difference of squares: \((g - 1)(g + 1)\).
So, the completely factored form is \(g(g - 1)(g + 1)(g^2 + 1)\). Since this is not the form given in the problem, this option was not factored completely.

  1. For \(4g^3 + 18g^2 + 20g\):

First, we extract the greatest common factor, \(2g\):
\[2g(2g^2 + 9g + 10)\]
Next, we factor the quadratic \(2g^2 + 9g + 10\). We look for factors of \(2 \times 10 = 20\) that add up to 9, which are 4 and 5.
\[2g^2 + 4g + 5g + 10 = 2g(g + 2) + 5(g + 2) = (2g + 5)(g + 2)\]
So, the completely factored form is \(2g(2g + 5)(g + 2)\). Since this is not the form given in the problem, this option was not factored completely.

  1. For \(24g^2 - 6g^4\):

First, we extract the greatest common factor, \(6g^2\):
\[6g^2(4 - g^2)\]
The expression \(4 - g^2\) is a difference of squares: \(2^2 - g^2 = (2 - g)(2 + g)\).
So, the completely factored form is \(6g^2(2 - g)(2 + g)\). Since this is not the form given in the problem, this option was not factored completely.

  1. For \(2g^2 + 5g + 4\):

To check if a quadratic \(ax^2 + bx + c\) can be factored, we look at its discriminant \(\Delta = b^2 - 4ac\).
Here, \(a = 2\), \(b = 5\), and \(c = 4\):
\[\Delta = 5^2 - 4(2)(4) = 25 - 32 = -7\]
Since the discriminant is negative, the quadratic has no real roots and cannot be factored further into linear factors with real coefficients. Thus, it is irreducible.

  1. Conclusion

Based on our analysis, the first three options were not presented in their completely factored form, as they could be broken down further. The fourth option, \(2g^2 + 5g + 4\), cannot be factored further because it is irreducible over the reals. Therefore, it is the only expression already "factored completely." The final answer is \(\boxed{2g^2 + 5g + 4}\).

Examples

In algebra, recognizing when a polynomial is completely factored is essential for simplifying complex rational expressions and solving equations. For example, if you are designing a structural component where the load distribution is modeled by a polynomial, factoring it helps you identify critical points where the stress might be zero or maximized. Just as we checked if \(2g^2 + 5g + 4\) could be reduced, engineers perform similar checks to simplify models, ensuring that calculations for bridge stability or machine parts are handled using the most fundamental, irreducible components.

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