Paul's Online Notes
Home / Algebra / Polynomial Functions / Zeroes/Roots of Polynomials
Show General Notice Show Mobile Notice Show All Notes Hide All Notes
General Notice

This is a little bit in advance, but I wanted to let everyone know that my servers will be undergoing some maintenance on May 17 and May 18 during 8:00 AM CST until 2:00 PM CST. Hopefully the only inconvenience will be the occasional “lost/broken” connection that should be fixed by simply reloading the page. Outside of that the maintenance should (fingers crossed) be pretty much “invisible” to everyone.

Paul
May 6, 2021

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.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

### Section 5-2 : Zeroes/Roots of Polynomials

For problems 1 – 6 list all of the zeros of the polynomial and give their multiplicities.

1. $$f\left( x \right) = {x^2} + 2x - 120$$
2. $$R\left( x \right) = {x^2} + 12x + 32$$
3. $$h\left( x \right) = 4{x^3} + {x^2} - 3x$$
4. $$A\left( x \right) = {x^5} + 2{x^4} - 35{x^3} + 92{x^2} - 92x + 32 = {\left( {x - 1} \right)^2}\left( {x + 8} \right){\left( {x - 2} \right)^2}$$
5. $$Q\left( x \right) = {x^{10}} + 17{x^9} + 115{x^8} + 387{x^7} + 648{x^6} + 432{x^5} = {x^5}{\left( {x + 3} \right)^3}{\left( {x + 4} \right)^2}$$
6. $$g\left( x \right) = {x^8} + 2{x^7} - 14{x^6} - 16{x^5} + 49{x^4} + 62{x^3} - 44{x^2} - 88x - 32 = \left( {x + 4} \right){\left( {x + 1} \right)^4}{\left( {x - 2} \right)^3}$$

For problems 7 – 11 $$x = r$$ is a root of the given polynomial. Find the other two roots and write the polynomial in fully factored form.

1. $$P\left( x \right) = {x^4} - 3{x^3} - 18{x^2}$$ ; $$r = 6$$
2. $$P\left( x \right) = {x^3} + {x^2} - 46x + 80$$ ; $$r = - 8$$
3. $$P\left( x \right) = {x^3} - 9{x^2} + 26x - 24$$ ; $$r = 3$$
4. $$P\left( x \right) = 12{x^3} + 13{x^2} - 1$$ ; $$r = - 1$$
5. $$P\left( x \right) = 4{x^3} + 11{x^2} - 134x - 105$$ ; $$r = 5$$

For problems 12 – 14 determine the smallest possible degree for a polynomial with the given zeros and their multiplicities.

1. $${r_1} = - 2$$ (multiplicity 1), $${r_2} = 1$$ (multiplicity 1), $${r_3} = 4$$ (multiplicity 1)
2. $${r_1} = 3$$ (multiplicity 4), $${r_2} = - 5$$ (multiplicity 1)
3. $${r_1} = 7$$ (multiplicity 2), $${r_2} = 4$$ (multiplicity 7), $${r_3} = - 10$$ (multiplicity 5)
4. A 7th degree polynomial has roots $${r_1} = - 9$$ (multiplicity 2) and $${r_{\,2}} = 3$$ (multiplicity 1). What is the maximum number of remaining roots for the polynomial?