1. PC - Functions Name __________________
How-to guide to long division with polynomials
We are goingto divide the factor 𝑥 + 1 intothe quadratic 𝑥2 − 9𝑥 − 10. Forthe moment,I'll
ignore the othertermsand lookjustat the leading 𝑥 of the divisorandthe leading 𝑥2 of the
dividend.
If I divide the leading 𝑥2 inside bythe leading 𝑥 infront,whatwouldIget? I'd get an 𝑥. So I'll
put an 𝑥 ontop:
NowI'll take that 𝑥, and multiplyitthroughthe divisor, 𝑥 + 1.First,I multiplythe 𝑥 (ontop)
by the 𝑥 (onthe "side"),andcarrythe 𝑥2 underneath.
ThenI'll multiplythe 𝑥 (ontop) bythe 1 (onthe "side"),andcarry the 1𝑥 underneath.
ThenI'll draw the "equals"bar,so I can do the subtraction. To subtractthe polynomials,
I changeall the signs in the secondline...
...andthenI add down.The firstterm(the 𝑥2) will cancel out
I needtoremembertocarry downthat last term, the "subtractten",from the dividend.
NowI lookat the 𝑥 fromthe divisorandthe newleadingterm, the – 10𝑥,inthe bottomline of
the division.If Idivide the – 10𝑥 bythe 𝑥,I wouldendupwitha – 10,so I'll put that on top.

2. NowI'll multiplythe – 10 (ontop) bythe leading 𝑥 (onthe "side"),and carrythe – 10𝑥 to the
bottom...
...andI'll multiplythe – 10 (ontop) bythe 1 (onthe "side"),andcarry the –10 to the bottom.
I draw the equalsbar,and changethesigns on all the termsin the bottomrow
ThenI add down.
Thenthe solutiontothisdivisionis: 𝒙 – 𝟏𝟎. Since the remainderonthisdivisionwaszero(thatis,since there wasn't
anythingleftover),the divisioncame out"even". Whenyoudoregulardivisionwithnumbersandthe divisioncomes
out even,itmeansthatthe numberyoudividedbyisa factor of the numberyou're dividing. Forinstance,if you
divide 50 by10, the answerwill be a nice whole "5"witha zeroremainder,because 10is a factor of 50. Inthe case of
the above polynomial division,the zeroremaindertellsusthat 𝑥 + 1 is a factor of 𝑥2– 9𝑥 – 10, whichyoucan confirm
by factoringthe original quadraticdividend, 𝑥2– 9𝑥 – 10.