Answer: g(x) = x^4 - 9x^3 + 18x^2 + 32x - 96 which is choice C
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Step-by-step explanation:
Given roots: -2, 4, 4, 3
Based on that, we know that x = -2, x = 4, x = 4, and x = 3. The repeat x value of 4 is needed to help deal with a double root (multiplicity 2)
x = -2 leads to x+2 = 0, so (x+2) is one factor
x = 4 leads to x-4 = 0, making (x-4) another factor. We have two copies of (x-4) as a factor
x = 3 leads to x-3 = 0 so (x-3) is the last factor
Overall, the four factors are: (x+2) and (x-4) and (x-4) and (x-3)
Use the distributive property to expand everything out
g(x) = (x+2)(x-4)(x-4)(x-3)
g(x) = ( (x+2)(x-4) ) * ( (x-4)(x-3) )
g(x) = ( x^2 - 2x - 8 ) * ( x^2 - 7x + 12 )
g(x) = x^2( x^2 - 7x + 12 ) - 2x( x^2 - 7x + 12 ) - 8( x^2 - 7x + 12 )
g(x) = x^4 - 7x^3 + 12x^2 -2x^3 + 14x^2 - 24x - 8x^2 + 56x - 96
g(x) = x^4 - 9x^3 + 18x^2 + 32x - 96
which shows how I got choice C as the answer