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

Q.

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

The function

f

is shown. If

x-3

is a factor of

f

, what is the value of

f(3)

Check Factor by Substitution: Since $x-3$ is a factor of $f(x)$ , we know that $f(3)$ should equal $0$ . This is because if $x-3$ is a factor, then $f(x)$ must be divisible by $x-3$ , and the remainder when $f(x)$ is divided by $x-3$ should be $0$ . We can verify this by substituting $f(x)$ $0$ into the function $f(x)$ and checking if the result is indeed $0$ .
Substitute $x=3$ : Substitute $x=3$ into the function $f(x)$ : $f(3) = 2(3)^5 + (3)^4 - 18(3)^3 - 17(3)^2 + 20(3) + 12$
Calculate $f(3)$ : Calculate the value of $f(3)$ : $f(3) = 2(243) + 81 - 18(27) - 17(9) + 60 + 12$
Continue Calculation: Continue the calculation: $f(3) = 486 + 81 - 486 - 153 + 60 + 12$
Simplify Expression: Simplify the expression: $f(3) = 486 - 486 + 81 - 153 + 60 + 12$
Combine Like Terms: Combine like terms: $f(3) = 0 + 81 - 153 + 60 + 12$
Finish Calculation: Finish the calculation: $f(3) = 0$
Confirm Factor: Since we have found that $f(3) = 0$ , this confirms that $x-3$ is indeed a factor of $f(x)$ , as expected. The value of $f(3)$ is $0$ , which is consistent with the fact that $x-3$ is a factor of the polynomial $f(x)$ .

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