When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other options? I know there is the difference of two cubes technique, factoring out an $x$ might work if possible(then factoring polynomial with $x^2$), or factoring by grouping.
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$\begingroup$Personally, when I am required to solve a polynomial with degree greater than 2, I like to use the Rational Root Theorem to find potential roots, and then use synthetic division using a root when it is found in order to reduce it to a nicer polynomial.
Note there is an explicit formula for cubic polynomials, but it is messy.
Using your question as an example, we know that possible rational roots are (plus or minus) factors of $48$. Just from guessing, we know that $-2$ is a root. Therefore we can reduce to the quadratic $-x^2 - 2x + 24$, which can be easily solved using the quadratic equation.
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