Algebra 2
Unit Activity
Unit: Polynomial, Rational, and Radical Relationships
This Unit Activity will help you meet these educational goals:
 Mathematical Practices—You will make sense of problems and solve them, use
mathematics to model real-world situations, and use appropriate tools strategically.
 Inquiry—You will conduct online research in which you will collect information and
communicate results in tables, graphs, and written form.
Introduction
You probably have known how to multiply numbers for a long time. For one-digit numbers,
memorization is usually the technique used to multiply them. While you know that 3 × 3 really
means 3 + 3 + 3, you probably don’t bother to add three to itself three times; you just know
that 3 × 3 = 9. For numbers with two or more digits, you memorized a process that lets you
arrive at the correct answer rather than memorizing every possible multiplication problem.
Have you ever thought that the process we use to multiply numbers seems arbitrary? Have
you ever wondered why it works? You can use polynomials to understand why it works, and
you can use polynomial rational expressions to better understand division too.
__________________________________________________________________________
Directions and Analysis
Task 1: Numbers as Polynomials
The first step to understanding why multiplication works is understanding what a multiple-digit
integer really is. You have been working with integers for so long that this understanding is
intuitive. In the decimal system, each digit represents a different power of ten. The integer 1
is equal to 100, the integer 10 is equal to 101, the integer 100 is equal to 102, the integer
1,000 equals 103, and so on.
a. All of the examples in the above paragraph refer to integers that include only the digit 1.
What would the integer 500 equal using powers of 10? How did you come up with that
answer?
Type your response here:
b. Integers with different values in each digit can be expanded in a similar way. For these
integers, you must use addition between each digit. In the case of 24, this means using
the form (2 × 101) + (4 × 100). What would the integer 2,341 look like if you used this
notation?
Type your response here:
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1
c. Looking at your response to part b, how are these expansions of numbers similar to
polynomials?
Type your response here:
d. When adding two numbers, such as 123 and 423, care is taken to first line them up and
then add like digits. How does expanding this expression to [(1 × 102) + (2 × 101) + (3 ×
100)] + [(4 × 102) + (2 × 101) + (3 × 100)] make the operation more like a polynomial
addition problem?
Type your response here:
Task 2: Operations on Integers as Polynomials
Now that you have explored the similarities between integer multiplication and polynomial
multiplication, you will examine the similarities between integer long division and polynomial
division long division.
a. When multiplying the two integers 21 and 12, the following process is used:
21
× 12
Set up the problem.
21
× 12
42
Multiply the 2 of the 12 by 21.
21
× 12
42
210
Place a 0 below the 42, then multiply the 1 of the 12 by 21.
21
× 12
42
+ 210
252
Add 42 and 210 to get the answer: 252.
Expanding each integer in the expression 21 × 12, the result is [(2 × 101) + (1 × 100)] × [(1
× 101) + (2 × 100)]. Replace the 10s with x’s and then multiply this expression using FOIL.
Compare it with the normal integer multiplication method used above: How are they
similar? Show your work.
Type your response here:
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b. Why does using FOIL on polynomial expressions match so closely to integer
multiplication? Look closely at how the usual method works.
Type your response here:
c. The integer 844 can be rewritten as (8 × 102) + (4 × 101) + (4 × 100). If you replace 10 with
x, this integer can be written in the form 8x2 + 4x + 4. Use long division to divide 844 by 4.
How is the answer you get related to dividing the polynomial 8x2 + 4x + 4 by 4?
Type your response here:
d. Despite polynomial division and integer division being very similar, there are some
differences. The expression 12 × 36 is an example. Letting x = 10, (x + 2)(3x + 6) is an
equivalent expression. Multiply this using FOIL. In this case the coefficients are not simply
the digits of the answer. Why? Does this mean that FOIL gave a wrong answer?
Type your response here:
e. A rational number is a number that can be expressed as a fraction with both the
numerator and denominator being integers. How are rational expressions related to
rational numbers?
Type your response here:
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Numbers as Polynomials
a. Expanding an integer into powers of 10
Task points: 4
 1
b. Expanding a 4-digit integer into powers of 10

1
c. Relating integer expansion to polynomials

1
d. Polynomial addition

1
3
Task 2: Operations on Integers as Polynomials
a. Using FOIL on integer expansion
Task points: 6
 1
b. Comparing FOIL with integer multiplication

1
c. Comparing polynomial division with integer division

1
d. Examining a case where FOIL yields a different result

1
e. Describing the relationship between rational integers and
rational expressions

2
4