Paul's Online Notes
Paul's Online Notes
Home / Algebra / Preliminaries / Factoring Polynomials
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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 Solution

Don’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 2

Now, 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 3

Finally, once we write down the factors of the -28 we can see that the factoring of this polynomial is,

\[3{z^5} - 17{z^4} - 28{z^3} = \require{bbox} \bbox[2pt,border:1px solid black]{{{z^3}\left( {3z + 4} \right)\left( {z - 7} \right)}}\]