Answer :
Final Answer:
The factor of the polynomial [tex]\(x^5 - 3x^4 - 23x^3 + 51x^2 + 94x - 120\) is \((x - 5)\).[/tex]
Explanation:
The given polynomial is [tex]\(x^5 - 3x^4 - 23x^3 + 51x^2 + 94x - 120\)[/tex]. To find its factor, we need to look for a value of x that makes the polynomial equal to zero, which indicates that [tex]\(x - a\)[/tex] is a factor if a is that particular value of x.
Using synthetic division or substitution, we find that when [tex]\(x = 5\)[/tex], the polynomial becomes:
[tex]\[5^5 - 3 \cdot 5^4 - 23 \cdot 5^3 + 51 \cdot 5^2 + 94 \cdot 5 - 120 = 0.\][/tex]
This confirms that \(x - 5\) is a factor of the polynomial. The division process involves substituting [tex]\(x = 5\)[/tex] into the polynomial and carrying out the calculations step by step, which validates that the remainder is zero.
In the context of polynomial division, when the remainder is zero after dividing the polynomial by a linear factor, that factor is indeed a factor of the polynomial. This is because the remainder indicates how far off the polynomial is from being perfectly divisible by the factor. If the remainder is zero, there is no discrepancy, and the factor is valid.
In this case, [tex]\(x - 5\)[/tex] is the factor that satisfies this condition.
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