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The function f(x)=2x5+x4−18x3−17x2+20x+12 f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12 is shown. If x−3 x - 3 is a factor of f f , what is the value of f(3) f(3) ?

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Q. The function f(x)=2x5+x4−18x3−17x2+20x+12 f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12 is shown. If x−3 x - 3 is a factor of f f , what is the value of f(3) f(3) ?
  1. Understand Relationship: Understand the relationship between factors and function values.\newlineIf x−3x - 3 is a factor of the function f(x)f(x), then f(3)f(3) must be equal to 00, because when x=3x = 3, the factor x−3x - 3 becomes zero, and anything multiplied by zero is zero.
  2. Substitute x=3x = 3: Substitute x=3x = 3 into the function f(x)f(x) to find f(3)f(3).
    f(x)=2x5+x4−18x3−17x2+20x+12f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12
    f(3)=2(3)5+(3)4−18(3)3−17(3)2+20(3)+12f(3) = 2(3)^5 + (3)^4 - 18(3)^3 - 17(3)^2 + 20(3) + 12
  3. Calculate f(3)f(3): Calculate the value of f(3)f(3).
    f(3)=2(243)+(81)−18(27)−17(9)+20(3)+12f(3) = 2(243) + (81) - 18(27) - 17(9) + 20(3) + 12
    f(3)=486+81−486−153+60+12f(3) = 486 + 81 - 486 - 153 + 60 + 12
  4. Simplify Expression: Simplify the expression to find the final value of f(3)f(3).f(3)=486+81−486−153+60+12f(3) = 486 + 81 - 486 - 153 + 60 + 12f(3)=0+81−153+60+12f(3) = 0 + 81 - 153 + 60 + 12f(3)=81−153+60+12f(3) = 81 - 153 + 60 + 12f(3)=−72+60+12f(3) = -72 + 60 + 12f(3)=−12+12f(3) = -12 + 12f(3)=0f(3) = 0

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