Section 1.5 : Factoring Polynomials
17. Factor the following polynomial.
\[3{z^5} - 17{z^4} - 28{z^3}\]Show All Steps Hide All Steps
Start SolutionDon’t let the fact that this polynomial is not quadratic worry you. Just because it’s not a quadratic polynomial doesn’t mean that we can’t factor it.
For this polynomial note that we can factor a \({z^3}\) out of each term to get,
\[3{z^5} - 17{z^4} - 28{z^3} = {z^3}\left( {3{z^2} - 17z - 28} \right)\] Show Step 2Now, notice that the second factor is a quadratic and we know how to factor these. So, it looks like the form of the factoring should be,
\[3{z^5} - 17{z^4} - 28{z^3} = {z^3}\left( {3z + \underline {\,\,\,\,} } \right)\left( {z + \underline {\,\,\,\,} } \right)\] Show Step 3Finally, once we write down the factors of the -28 we can see that the factoring of this polynomial is,
\[3{z^5} - 17{z^4} - 28{z^3} = \require{bbox} \bbox[2pt,border:1px solid black]{{{z^3}\left( {3z + 4} \right)\left( {z - 7} \right)}}\]