Dividing polynomial p(x) by g(x) (with g(x)=0) always produces a quotient and remainder satisfying:
p(x)=g(x)q(x)+r(x),r(x)=0 or degr(x)<degg(x)
Worked example
Divide p(x)=x3−3x2+5x−3 by g(x)=x2−2.
Solution:x3 divided by x2 gives x; subtracting x(x2−2)=x3−2x leaves −3x2+7x−3. Then −3x2 divided by x2 gives −3; subtracting −3(x2−2)=−3x2+6 leaves 7x−9, whose degree is now below the divisor's:
Quotient=x−3,Remainder=7x−9
Worked example: finding remaining zeroes
Two zeroes of 3x4+6x3−2x2−10x−5 are ±5/3. Find the other zeroes.
Solution: Since ±5/3 are zeroes, (x−5/3)(x+5/3)=x2−35 is a factor. Dividing the original polynomial by this factor gives quotient 3x2+6x+3, which factors as:
3x2+6x+3=3(x+1)2
So the remaining two zeroes are both -1 (a repeated root).
Continue learning
Dividing out a known factor to uncover the remaining zeroes only works because of the same sum/product relationship between a polynomial's zeroes and its coefficients.