Polynomial Products

Task: Predict total degree of a product or degrees in which nonzero terms will appear. Rewrite a product of expressions as a polynomial in a specified variable.
Teaching notes: products of sums, Separation of tasks, diagnostic aids, Symbols in coefficients, No FOIL

Notation

We say an expression is written as a polynomial in a variable if it is explicitly given as a linear combination of powers of the variable times coefficients independent of the variable. This allows us to formulate problems as “Rewrite the following expression as a polynomial in $y$ .”

Coefficients can either precede or follow the variable: either $(1+a)y^2$ or $y^2(1+a)$ . The first is traditional but the second is often more convenient when collecting terms etc.
Terms can be written with exponents either increasing or decreasing: either $ax^2+bx+c$ or $c+bx+ax^2$ . Both are used in practice and by computer algebra systems.

Degrees

Problem 1

When $(3 u^2+ a u+5)(u^{10}+2u^4-b u^2)$ is written as a polynomial in $u$ ,

what is the highest degree?
in which degrees are there nonzero coefficients? (alt: for which powers of $u$ are the terms nonzero?)

Terms occur in degrees 2, 3, 4, 5, 6, 10, 11, 12 and the total (largest) degree is 12. This problem reenforces the structure-exploiting scanning procedure described in products of sums with almost no arithmetic. Qualitative information of this sort is used to focus or avoid quantitative work.

Coefficients

Problem 1

Find the $y^3$ term in $(3 y^3 -y^2+5ay)(y^{2}+2y-a)$ when it is written as a polynomial in $y$ . (alt: find the coefficient on $y^3$ ).

This term comes from products of a summand from the first factor and a summand from the second that give total $y$ exponent 3. The $y^3$ is fixed and can be factored out ahead of time. Scanning and picking out the terms of interest gives

(1)

$y^3\bigr[(3)(-a) + (-1)(2)+(5a)(1)\bigl]$

Show notes

Problem 2

$(2+3\sqrt3)(7-5\sqrt3)$ can be written in the form $a+b\sqrt3$ with $a, b$ rational. Find $b$ .
Find the imaginary part of $(2+3i)(7-5i)$ .

These can be thought of as the polynomial product $(2+3x)(7-5x)$ , either evaluated at particular values of $x$ or with the variable required to satisfy an identity ( $x^2=3$ or $x^2=-1$ ). The identities take the degree-2 term down to degree 0 so the answer is the coefficient on the degree 1 terms. Pick these out as in problem 1.

Problem 3

When $(t^2+3at-5)(3t^2 -t+a)$ is written as a polynomial in $t$ it has $t^2$ term $-4 t^2$ . Find $a$ .

This is a two-step problem done by finding the $t^2$ coefficient as a symbolic expression, then solving the resulting equation for $a$ . This example has three terms in the coefficient and a linear equation in $a$ . Higher-order equations can be used when students have appropriate skills.

Full products

Problem 1

Write $(3 z^2 +5a)(z^{4}+2z^2-a)$ as a polynomial in $z$ .

Finding the solution divides into organizational and arithmetic phases. It is much easier when abstract structure is used to get as much as possible from the organizational phase.

Scanning degrees shows there will be terms in degree 0, 2, 4, and 6. Write

(2)

$z^6( \dots)+z^4(\dots)+z^2(\dots)+(\dots)$

with plenty of room between parentheses. Next scan through all ways of pairing one term from each factor and record coefficients in the appropriate place (without performing any arithmetic. This gives

(3)

$z^6((3)(1))+z^4((3)(2)+(5a)(1))+z^2((3)(-a)+(5a)(2))+((5a)(-a))$

The arithmetic phase consists of simplifying the coefficients. See Products of Sums for an abstraction of this process.

Show notes

Problem 2

Write $(a + 2 x + \frac13 x^2)(4+a - x + 2x^2)$ as a polynomial in $x$ .

Problems of this length are necessary to illustrate important topics and they do come up in later work. It is not necessary to have a lot of drill at this level if techniques are designed so that students can handle them when they arise. See Products of Sums

Problem 3

Use a calculator to exactly multiply $792534812076$ and $80134263$ .

These numbers are too large to multiply directly on calculators. Instead divide into 4-digit blocks and think of them as polynomials in $x$ with 4-digit coefficients, evaluated at $x=10^4$ . The product becomes

(4)

$(7925x^2+3481x+2076)(8013x+4263)$

Expand as in problem 1. Numerical processing of the coefficients can be done exactly on a calculator since they will be numbers with 9 or fewer digits. Finally evaluate at $x=10^4$ . This final step involves adding a list of large numbers with at most 3 nonzero digits in each column. This can be done with calculators with some care about carrying.

Show notes

Multiple products

The basic idea behind products of sums is not limited to two factors.

Problem 1

When $(3x^2 +2x +b)(x+5)(4x^4-x+1)$ is written as a polynomial in $x$ ,

in which degrees does it have nonzero coefficients?
what is the coefficient on $x$ ?

This is done by scanning through three-fold products obtained by taking a term from each factor. When the scanning process is mastered (e.g. by using fingers to track where one is in the process) this is an easy problem.

Instances of the binomial theorem

Find the $u^4$ term in $(2+7u^2)^5$ .

This is immediate from the binomial theorem. It can be done directly as follows:

We have five copies of $2+7u^2$ . We get $u^4$ terms by choosing two copies from which to take $7u^2$ and taking $2$ from the other three. Each such choice contributes $7^2 \times 2^3$ to the coefficient. The coefficient is therefore the product of this and the number of ways there are to choose two things from five possible.

If students have been introduced to binomial coefficients in combinatorics or probability classes this gives a powerful application and connects the subjects. Otherwise this coefficient can be done directly. This is awkward enough to make development of binomial coefficients look powerful and useful.

The binomial theorem itself is more pedagogically advanced than the material above since both binomial coefficients and summation notation are required.

page tags: degree polynomial product

AMS Technical Careers

A K-12 Education Project

Notation

Degrees

Problem 1

Coefficients

Problem 1

Problem 2

Problem 3

Full products

Problem 1

Problem 2

Problem 3

Multiple products

Problem 1

Instances of the binomial theorem