Use the remainder theorem to evaluate the polynomial for the given values of \( x \). \[ f(x)=3 x^{3}-x^{2}-18 x+6 \]
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To evaluate the polynomial \( f(x) = 3x^{3} - x^{2} - 18x + 6 \) using the remainder theorem, you need to find \( f(a) \) for a specific value of \( a \). The remainder theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - a \) is \( f(a) \). For example, let’s evaluate \( f(2) \): \[ f(2) = 3(2)^{3} - (2)^{2} - 18(2) + 6 \] \[ = 3(8) - 4 - 36 + 6 \] \[ = 24 - 4 - 36 + 6 \] \[ = 24 - 34 \] \[ = -10 \] So, \( f(2) \) gives us a remainder of \( -10 \). You can replace 2 with any other value of x you'd like to evaluate!