Polynomial Division

Tasks: Find quotient and remainder in problems stated in both fraction and product form. For linear divisors iterate to rewrite as a polynomial in the divisor. This can also be described as finding “digits” with respect to a (linear) polynomial base.
Basic method: Long (synthetic) division.
Teaching notes: Diagnostic Aids, Symbols.

Notation

Terminology should reflect the fact that the methods exploit the same underlying mathematical structures as integer fractions and long division.

In particular a fraction $P/S$ is proper if the top is smaller than the bottom, with “smaller” referring to absolute value in the integer case, and to degree for polynomials.

There are two formats for expressing the result of division:

(whole thing plus proper fraction) $P/S = Q + R/S$ with $R/S$ a proper fraction; or
(factored plus remainder) $P = QS + R$ with $R$ smaller than $S$ .

Students should realize these are the same and be able to move back and fourth, with the exception that when discussing roots the factored form must be used to avoid division by zero.

Division

The standard presentation of the division algorithm is not problematic and does not need review.

Problem 1

Express $\frac{3 y^3-y^2-3}{y-2}$ as a polynomial plus a proper fraction.
Alternate: Express $3 y^5-y^2-3$ as a polynomial times ${y-2}$ , plus remainder.

Answer formats are $3 y^2+5 y+10+\frac{17}{y-2}$ and $(3 y^2+5 y+10)(y-2)\,+\,(17)$ respectively.

As noted above, students should see these as different forms of the same problem.

Problem 2

Find the remainder of $w^4-2w^2+\frac34$ after division by $w^2-w+3$

Problem 3

Does $u^2+3$ divide exactly into $2u^3-u^2+6u+3$ ?

Problem 4

Find $b$ so that $z^2-1$ divides exactly into $z^3-\frac32 z^2+b$ .

Linear Remainders

Important Fact: The remainder after dividing $P(x)$ by $x-r$ is $P(r)$ .

This — and the corollary below — is important enough that it is worth explaining why it is true.

Show Reason

Problem 1

Find the remainder of $w^4-2w^2+\frac34$ after division by $w+3$
Does $u+3$ divide exactly into $2u^3-u^2+6u+3$ ?
Find $b$ so that $z-2$ divides exactly into $z^3-\frac32 z^2+b$ .

Caution: Here $w+3$ should be thought of as $w-(-3)$ and it may reduce the error rate for students to write this explicitly.

Problem 2

Find the remainder of $2 x^3-ax +1$ after division by $2x+a$ , when these are considered as polynomial in $x$ .
Find the remainder when these are considered as polynomials in $a$ .

Symbols in coefficients are common and important (see Symbols). This also illustrates that an expression can be a polynomial with respect to several symbols, and the traditional $x$ may not be the one of interest.

Change of variable

A polynomial in $x$ can be rewritten as a polynomial in $y=a\,x +b$ provided $a\neq 0$ . This can be done by repeated division: $P(x)=(ax+b)Q_1(x) + R_1 =(ax+b)\bigl((ax+b)Q_2 +R_2\bigr)+R_1=\dots$ so the sequence of remainders are the new coefficients. Alternatively it can be done by solving $y=a\,x +b$ for $x$ , substituting in, and sorting out the mess. The division approach has the advantage that it extends to a version for higher-order dependent variables but this is not explored here.

Caution: if a new symbol ( $y$ in the above) is introduced then the relation to the old variable ( $y=a\,x +b$ in the above) is part of the answer. The transformed expression alone is incomplete and an invitation to errors.

Note: If students have seen integers expressed in bases different than 10 then it can be pointed out that the same thing is going on here. The coefficients in a polynomial correspond to digits in a number. Coefficients in the rewritten polynomial are therefore the “base $a\,x +b$ digits” for the polynomial.

Problem 1

Rewrite $3y^2-2y+5$ as a polynomial in $x=y-2$ .

(Or, find the base $y-2$ digits of $3y^2-2y+5$ .)

Answer: either of:

$3(y-2)^2+10(y-2)+13$
$3x^2+10x+13$ where $x =y-2$ .

The result of the first division is $3y+4+\frac{13}{y-2}$ , and this would make a good diagnostic aid (see Diagnostic Aids.

Problem 2

Write $2 x^3-ax +1$ as a polynomial in $y=2x+a$ when these are considered polynomials in $x$ ).
Write $2 x^3-ax +1$ as a polynomial in $y=2x+a$ when these are considered polynomials in $a$ ).

Note: Changing variables requires completely replacing the old variable. Symbols other than the one specified as the “variable” will generally still appear, but an appearance of the old variable is an ‘error message’ since it means the answer must be wrong.

Problem 3

$-2$ is a root of $u^4+2u^3+8u^2+13u -6$ . Change variables to get a polynomial with no constant term.

Since $-2$ is a root the change of variables $z=u-2$ will have no remainder in the first division and so no constant term.

Answer: $z^4-6z^3+20z^2-27z$ , where $z=u-2$ .

Page tags: changevariables fraction polynomial quotient remainder

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Notation

Division

Problem 1

Problem 2

Problem 3

Problem 4

Linear Remainders

Problem 1

Problem 2

Change of variable

Problem 1

Problem 2

Problem 3