The function f(x) = ax^3 - x^2 + bx - 24 has three factors. Two of these factors are x - 2 and x + 4. Determine the values of a and b.

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llltkl | College Teacher | (Level 3) Valedictorian

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Two of the factors of the function `f(x) = ax^3 - x^2 + bx - 24` are (x - 2) and (x + 4).

According to the factor theorem, f(2) and f(-4) must be equal to zero.

So, `a*2^3 - 2^2 + b*2 - 24=0`   --- (i)

`rArr 8a+2b-28=0`

And, `a(-4)^3 - (-4)^2 + b(-4) - 24=0`

`rArr 16a+b+10=0` --- (ii)

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2*(ii)-(i)

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`24a+48=0`

`rArr a=-2`

Put this vaue in (i),

`8*(-2)+2b=28`

`rArr 2b=28+16=44`

`rArr b=22`

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