Question

How do you use the factor theorem to determine whether x-5 is a factor of `3x^3 - 12x^2 - 11x - 20`?

Answer 1

Since `f(5) = 0`
`color(white)("XXX")`(for `f(x)=3x^3-12x^2-11x-20`)
`(x-5)` is a factor of `(f(x)`

Explanation:

The Factor Theorem says that `(x-a)` is a factor of `f(x)` if and only if `f(a)=0`

You could use a calculator, synthetic substitution (below), paper and pencil, or many other ways to evaluate `3x^3-12x^2-11x-20` at `x=5`

Evaluation using synthetic substitution:
#{: (,,3,-12,-11,-20), (,,,color(white)("X")15,color(white)("X")15,color(white)("X")20), (,,"----","-----","-----","-----"), (xx(5),"|",3,color(white)("X")3,color(white)("X")4,color(white)("X")color(red)(0)) :}#