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Answer :
To prove that [tex]x - 2[/tex] is a factor of the polynomial [tex]f(x) = 2x^3 - 23x^2 + 80x - 84[/tex], we can use the Factor Theorem. The Factor Theorem states that [tex]x - c[/tex] is a factor of a polynomial [tex]f(x)[/tex] if and only if [tex]f(c) = 0[/tex].
Here’s how we can apply this theorem step-by-step:
Identify [tex]c[/tex] from the Factor: Since we are given [tex]x - 2[/tex], we have [tex]c = 2[/tex].
Substitute [tex]c[/tex] into the polynomial: We will substitute [tex]2[/tex] into the polynomial [tex]f(x)[/tex] to find [tex]f(2)[/tex].
[tex]f(2) = 2(2)^3 - 23(2)^2 + 80(2) - 84[/tex]
Calculate [tex]f(2)[/tex]:
[tex]f(2) = 2(8) - 23(4) + 160 - 84[/tex]
[tex]f(2) = 16 - 92 + 160 - 84[/tex]
[tex]f(2) = 0[/tex]Conclusion: Since [tex]f(2) = 0[/tex], by the Factor Theorem, [tex]x - 2[/tex] is a factor of [tex]f(x)[/tex].
Thus, we have proven that [tex]x - 2[/tex] is indeed a factor of the given polynomial.
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