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 and then solve for $f(x)$. Please give an algebraic solution.
2026-04-01 14:52:53.1775055173
Factor Theorem given two factors
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HINT : By the factor theorem, we have $$F(2)=a\cdot 2^3-2^2+b\cdot 2-24=0\tag1$$$$F(-4)=a\cdot (-4)^3-(-4)^2+b\cdot(-4)-24=0\tag2$$ Now, you can solve these for $a,b$.
$$(1)\Rightarrow 8a+2b=28\Rightarrow 4a+b=14.$$ $$(2)\Rightarrow -64a-4b=40\Rightarrow -16a-b=10.$$