Use the remainder theorem to find the remainder when \( f(x) \) is divided by \( x+2 \). Then use the factor theorem to determine whether \( x+2 \) is a factor of \( f(x) \). \( f(x)=2 x^{6}-8 x^{4}+x^{3}-9 \) The remainder is

To find the remainder when \( f(x) \) is divided by \( x+2 \), we can use the remainder theorem, which states that the remainder of the division of \( f(x) \) by \( x - c \) is given by \( f(c) \). Here, since we are dividing by \( x + 2 \), we let \( c = -2 \). Now, substituting \( -2 \) into \( f(x) \): \[ f(-2) = 2(-2)^{6} - 8(-2)^{4} + (-2)^{3} - 9 \] Calculating each term: \[ 2(-2)^{6} = 2 \times 64 = 128 \] \[ -8(-2)^{4} = -8 \times 16 = -128 \] \[ (-2)^{3} = -8 \] \[ -9 = -9 \] Putting it all together: \[ f(-2) = 128 - 128 - 8 - 9 = -17 \] Thus, the remainder is \( -17 \). Now, to determine whether \( x + 2 \) is a factor of \( f(x) \), we can use the factor theorem. According to this theorem, \( x - c \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \). In this case, since we found \( f(-2) = -17 \), which is not equal to \( 0 \), we conclude that \( x + 2 \) is not a factor of \( f(x) \).