f(x)=2x^(5)+x^(4)-18x^(3)-17x^(2)+20 x+12 The function f is shown. If x-3 is a factor of f, what is the value of f(3) ?

Verify Factor Theorem: If x-3 is a factor of f(x), then by the Factor Theorem, f(3) should equal 0. Let's calculate f(3) to verify this. f(3) = 2(3)^5 + (3)^4 - 18(3)^3 - 17(3)^2 + 20(3) + 12Calculate f(3): Now, let's perform the calculations step by step. f(3) = 2(243) + 81 - 18(27) - 17(9) + 60 + 12Perform Step by Step Calculations: Simplify the terms. f(3) = 486 + 81 - 486 - 153 + 60 + 12Simplify Terms: Combine like terms. f(3) = 486 - 486 + 81 - 153 + 60 + 12Combine Like Terms: Further simplifying, we get: f(3) = 0 + 81 - 153 + 60 + 12Further Simplify: Finally, we add up the remaining terms. f(3) = 81 - 153 + 60 + 12 f(3) = -72 + 60 + 12 f(3) = -12 + 12 f(3) = 0

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f(x)=2x^(5)+x^(4)-18x^(3)-17x^(2)+20 x+12
The function
f is shown. If
x-3 is a factor of
f, what is the value of
f(3) ?

$f(x)=2 x^{5}+x^{4}-18 x^{3}-17 x^{2}+20 x+12$ $\newline$ The function $f$ is shown. If $x-3$ is a factor of $f$ , what is the value of $f(3)$ ?

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Full solution

f(x)=2 x^{5}+x^{4}-18 x^{3}-17 x^{2}+20 x+12

\newline

The function

f

is shown. If

x-3

is a factor of

f

, what is the value of

f(3)

Verify Factor Theorem: If $x-3$ is a factor of $f(x)$ , then by the Factor Theorem, $f(3)$ should equal $0$ . Let's calculate $f(3)$ to verify this. $f(3) = 2(3)^5 + (3)^4 - 18(3)^3 - 17(3)^2 + 20(3) + 12$
Calculate $f(3)$ : Now, let's perform the calculations step by step. $f(3) = 2(243) + 81 - 18(27) - 17(9) + 60 + 12$
Perform Step by Step Calculations: Simplify the terms. $\newline$ $f(3) = 486 + 81 - 486 - 153 + 60 + 12$
Simplify Terms: Combine like terms. $\newline$ $f(3) = 486 - 486 + 81 - 153 + 60 + 12$
Combine Like Terms: Further simplifying, we get: $\newline$ $f(3) = 0 + 81 - 153 + 60 + 12$
Further Simplify: Finally, we add up the remaining terms. $\newline$ $f(3) = 81 - 153 + 60 + 12$ $\newline$ $f(3) = -72 + 60 + 12$ $\newline$ $f(3) = -12 + 12$ $\newline$ $f(3) = 0$