lesson 8: adding and subtracting polynomials

M1 Lesson 8 ALGEBRA I

Lesson 8: Adding and Subtracting Polynomials

Student Outcomes

Students understand that the sum or difference of two polynomials produces another polynomial and relate polynomials to the system of integers; students add and subtract polynomials.

Classwork

Exercise 1 (7 minutes)

Have students complete Exercise 1(a) and use it for a brief discussion on the notion of base. Then have students continue with the remainder of the exercise.

Exercise 1

a. How many quarters, nickels, and pennies are needed to make $𝟏𝟏.𝟏𝟏𝟏𝟏?

Answers will vary.

𝟒𝟒 quarters, 𝟐𝟐 nickels, 𝟏𝟏 pennies

b. Fill in the blanks:

𝟖𝟖,𝟗𝟗𝟒𝟒𝟏𝟏 = 𝟖𝟖 × 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏+ 𝟗𝟗 × 𝟏𝟏𝟏𝟏𝟏𝟏+ 𝟒𝟒 × 𝟏𝟏𝟏𝟏 + 𝟏𝟏 × 𝟏𝟏 = 𝟖𝟖 × 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟗𝟗 × 𝟏𝟏𝟏𝟏𝟐𝟐 + 𝟒𝟒 × 𝟏𝟏𝟏𝟏+ 𝟏𝟏 × 𝟏𝟏

c. Fill in the blanks:

𝟖𝟖,𝟗𝟗𝟒𝟒𝟏𝟏 = 𝟏𝟏 × 𝟐𝟐𝟏𝟏𝟏𝟏 + 𝟐𝟐 × 𝟐𝟐𝟏𝟏𝟐𝟐 + 𝟕𝟕 × 𝟐𝟐𝟏𝟏 + 𝟏𝟏 × 𝟏𝟏

d. Fill in the blanks:

𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟒𝟒 × 𝟓𝟓𝟐𝟐 + 𝟐𝟐 × 𝟓𝟓 + 𝟏𝟏 × 𝟏𝟏

Next ask:

Why do we use base 10? Why do we humans have a predilection for the number 10?

Why do some cultures have base 20?

How do you say “87” in French? How does the Gettysburg address begin?

Quatre-vingt-sept: 4-20s and 7; Four score and seven years ago…

Computers use which base system? Base 2

Extension: Mayan, Aztec, and

Celtic all used base 20. The word “score” (which means 20) originated from the Celtic language.

Students could be asked to research more on this and on the cultures who use or used base 5 and base 60.


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In Exercise 2, we are laying the foundation that polynomials written in standard form are simply base 𝑥𝑥 “numbers.” The practice of filling in specific values for 𝑥𝑥 and finding the resulting values lays a foundation for connecting this algebra of polynomial expressions with the later lessons on polynomial functions (and other functions) and their inputs and outputs.

Work through Exercise 2 with the class.

Exercise 2

Now let’s be as general as possible by not identifying which base we are in. Just call the base 𝒙𝒙.

Consider the expression 𝟏𝟏 × 𝒙𝒙𝟏𝟏 + 𝟐𝟐 × 𝒙𝒙𝟐𝟐 + 𝟕𝟕 × 𝒙𝒙 + 𝟏𝟏 × 𝟏𝟏, or equivalently 𝒙𝒙𝟏𝟏 + 𝟐𝟐𝒙𝒙𝟐𝟐 + 𝟕𝟕𝒙𝒙 + 𝟏𝟏.

a. What is the value of this expression if 𝒙𝒙 = 𝟏𝟏𝟏𝟏?

𝟏𝟏,𝟐𝟐𝟕𝟕𝟏𝟏

b. What is the value of this expression if 𝒙𝒙 = 𝟐𝟐𝟏𝟏?

𝟖𝟖,𝟗𝟗𝟒𝟒𝟏𝟏

Point out that the expression we see here is just the generalized form of their answer from part (b) of Exercise 1. However, as we change 𝑥𝑥, we get a different number each time.


Allow students time to complete Exercise 3 individually. Then elicit responses from the class.

Exercise 3

a. When writing numbers in base 𝟏𝟏𝟏𝟏, we only allow coefficients of 𝟏𝟏 through 𝟗𝟗. Why is that?

Once you get ten of a given unit, you also have one of the unit to the left of that.

b. What is the value of 𝟐𝟐𝟐𝟐𝒙𝒙 + 𝟏𝟏 when 𝒙𝒙 = 𝟓𝟓? How much money is 𝟐𝟐𝟐𝟐 nickels and 𝟏𝟏 pennies?

𝟏𝟏𝟏𝟏𝟏𝟏

$𝟏𝟏.𝟏𝟏𝟏𝟏

c. What number is represented by 𝟒𝟒𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟕𝟕𝒙𝒙 + 𝟐𝟐 if 𝒙𝒙 = 𝟏𝟏𝟏𝟏?

𝟓𝟓𝟕𝟕𝟐𝟐

d. What number is represented by 𝟒𝟒𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟕𝟕𝒙𝒙 + 𝟐𝟐 if 𝒙𝒙 = −𝟐𝟐 or if 𝒙𝒙 = 𝟐𝟐𝟏𝟏 ?

−𝟏𝟏𝟏𝟏

𝟏𝟏𝟏𝟏𝟏𝟏𝟗𝟗

e. What number is represented by −𝟏𝟏𝒙𝒙𝟐𝟐 + √𝟐𝟐𝒙𝒙 + 𝟏𝟏𝟐𝟐 when 𝒙𝒙 = √𝟐𝟐?

−𝟕𝟕𝟐𝟐


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Point out, as highlighted by Exercises 1 and 3, that carrying is not necessary in this type of expression (polynomial expressions). For example, 4𝑥𝑥2 + 17𝑥𝑥 + 2 is a valid expression. However, in base ten arithmetic, coefficients of value ten or greater are not conventional notation. Setting 𝑥𝑥 = 10 in 4𝑥𝑥2 + 17𝑥𝑥 + 2 yields 4 hundreds, 17 tens, and 2 ones, which is to be expressed as 5 hundreds, 7 tens, and 2 ones.

Discussion (11 minutes)

The next item in your student materials is a definition for a polynomial expression. Read the definition carefully, and then create 3 polynomial expressions using the given definition.

Polynomial Expression: A polynomial expression is either

1. A numerical expression or a variable symbol, or

2. The result of placing two previously generated polynomial expressions into the blanks of the addition operator (__+__) or the multiplication operator (__×__).

Compare your polynomial expressions with a neighbor’s. Do your neighbor’s expressions fall into the category of polynomial expressions?

Resolve any debates as to whether a given expression is indeed a polynomial expression by referring back to the definition and discussing as a class.

Note that the definition of a polynomial expression includes subtraction (add the additive inverse instead), dividing by a non-zero number (multiply by the multiplicative inverse instead), and even exponentiation by a non-negative integer (use the multiplication operator repeatedly on the same numerical or variable symbol).

List several of the student-generated polynomials on the board. Include some that contain more than one variable.

Initiate the following discussion, presenting expressions on the board when relevant.

Just as the expression (3 + 4) ∙ 5 is a numerical expression but not a number, (𝑥𝑥 + 5) + (2𝑥𝑥2 − 𝑥𝑥)(3𝑥𝑥 + 1) is a polynomial expression but not technically a polynomial. We reserve the word polynomial for polynomial expressions that are written simply as a sum of monomial terms. This begs the question: What is a monomial?

A monomial is a polynomial expression generated using only the multiplication operator (__×__). Thus, it does not contain + or − operators.

Just as we would not typically write a number in factored form and still refer to it as a number (we might call it a number in factored form), similarly, we do not write a monomial in factored form and still refer to it as a monomial. We multiply any numerical factors together and condense multiple instances of a variable factor using (whole number) exponents.

Try creating a monomial.

Compare the monomial you created with your neighbor’s. Is your neighbor’s expression really a monomial? Is it written in the standard form we use for monomials?

There are also such things as binomials and trinomials. Can anyone make a conjecture about what a binomial is and what a trinomial is and how they are the same or different from a polynomial?


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Students may conjecture that a binomial has two of something and that a trinomial three of something. Further, they might conjecture that a polynomial has many of something. Allow for discussion and then state the following:

A binomial is the sum (or difference) of two monomials. A trinomial is the sum (or difference) of three monomials. A polynomial, as stated earlier, is the sum of one or more monomials.

The degree of a monomial is the sum of the exponents of the variable symbols that appear in the monomial.

The degree of a polynomial is the degree of the monomial term with the highest degree.

While polynomials can contain multiple variable symbols, most of our work with polynomials will be with polynomials in one variable.

What do polynomial expressions in one variable look like? Create a polynomial expression in one variable and compare with your neighbor.

Post some of the student generated polynomials in one variable on the board.

Let’s relate polynomials to the work we did at the beginning of the lesson.

Is this expression an integer in base 10? 10(100 + 22 − 2) + 3(10) + 8 − 2(2)

Is the expression equivalent to the integer 1,234? How did we find out?

We rewrote the first expression in our standard form, right?

Polynomials in one variable have a standard form as well. Use your intuition of what standard form of a polynomial might be to write this polynomial expression as a polynomial in standard form: 2𝑥𝑥(𝑥𝑥2 − 3𝑥𝑥 + 1) − (𝑥𝑥3 + 2) and compare your result with your neighbor.

Students should arrive at the answer 𝑥𝑥3 − 6𝑥𝑥2 + 2𝑥𝑥 − 2.

Confirm that in standard form, we start with the highest degreed monomial, and continue in descending order.

The leading term of a polynomial is the term of highest degree that would be written first if the polynomial is put into standard form. The leading coefficient is the coefficient of the leading term.

What would you imagine we mean when we refer to the constant term of the polynomial?

A constant term is any term with no variables. To find “the constant” term of a polynomial, be sure you have combined any and all constant terms into one single numerical term, written last if the polynomial is put into standard form. Note that a polynomial does not have to have a constant term (or could be said to have a constant term of 0).

As an extension for advanced students, assign the task of writing of a formal definition for standard form of a polynomial. The formal definition is provided below for your reference:

A polynomial expression with one variable symbol 𝑥𝑥 is in standard form if it is expressed as, 𝑎𝑎𝑛𝑛𝑥𝑥𝑛𝑛 +𝑎𝑎𝑛𝑛−1𝑥𝑥𝑛𝑛−1 + ⋯+ 𝑎𝑎1𝑥𝑥 + 𝑎𝑎0, where 𝑛𝑛 is a non-negative integer, and 𝑎𝑎0, 𝑎𝑎1, 𝑎𝑎2, … , 𝑎𝑎𝑛𝑛 are constant coefficients with 𝑎𝑎𝑛𝑛 ≠ 0. A polynomial expression in 𝑥𝑥 that is in standard form is often called a polynomial in 𝑥𝑥.


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Exercise 4 (5 minutes) Exercise 4

Find each sum or difference by combining the parts that are alike.

a. 𝟒𝟒𝟏𝟏𝟕𝟕+ 𝟐𝟐𝟏𝟏𝟏𝟏 = 𝟒𝟒 hundreds + 𝟏𝟏 tens + 𝟕𝟕 ones + 𝟐𝟐 hundreds + 𝟏𝟏 tens + 𝟏𝟏 ones

= 𝟏𝟏 hundreds + 𝟒𝟒 tens + 𝟖𝟖 ones

b. (𝟒𝟒𝒙𝒙𝟐𝟐 + 𝒙𝒙 + 𝟕𝟕) + (𝟐𝟐𝒙𝒙𝟐𝟐 + 𝟏𝟏𝒙𝒙 + 𝟏𝟏)

𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟒𝟒𝒙𝒙 + 𝟖𝟖

c. (𝟏𝟏𝒙𝒙𝟏𝟏 − 𝒙𝒙𝟐𝟐 + 𝟖𝟖) − (𝒙𝒙𝟏𝟏 + 𝟓𝟓𝒙𝒙𝟐𝟐 + 𝟒𝟒𝒙𝒙− 𝟕𝟕)

𝟐𝟐𝒙𝒙𝟏𝟏 − 𝟏𝟏𝒙𝒙𝟐𝟐 − 𝟒𝟒𝒙𝒙 + 𝟏𝟏𝟓𝟓

d. 𝟏𝟏(𝒙𝒙𝟏𝟏 + 𝟖𝟖𝒙𝒙) − 𝟐𝟐(𝒙𝒙𝟏𝟏 + 𝟏𝟏𝟐𝟐)

𝒙𝒙𝟏𝟏 + 𝟐𝟐𝟒𝟒𝒙𝒙 − 𝟐𝟐𝟒𝟒

e. (𝟓𝟓 − 𝒕𝒕 − 𝒕𝒕𝟐𝟐) + (𝟗𝟗𝒕𝒕 + 𝒕𝒕𝟐𝟐)

𝟖𝟖𝒕𝒕+ 𝟓𝟓

f. (𝟏𝟏𝒑𝒑 + 𝟏𝟏) + 𝟏𝟏(𝒑𝒑 − 𝟖𝟖) − (𝒑𝒑 + 𝟐𝟐)

𝟖𝟖𝒑𝒑 − 𝟒𝟒𝟗𝟗

Closing (3 minutes)

How are polynomials analogous to integers?

While integers are in base 10, polynomials are in base 𝑥𝑥.

If you add two polynomials together, is the result sure to be another polynomial? The difference of two polynomials?

Students will likely reply, “yes,” based on the few examples and their intuition. Are you sure? Can you think of an example where adding or subtracting two polynomials does not result in a

polynomial?

Students thinking about 𝑥𝑥2 − 𝑥𝑥2 = 0 could suggest not. At this point, review the definition of a polynomial. Constant symbols are polynomials.


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Exit Ticket (4 minutes)

Lesson Summary

A monomial is a polynomial expression generated using only the multiplication operator (__×__). Thus, it does not contain + or − operators. Monomials are written with numerical factors multiplied together and variable or other symbols each occurring one time (using exponents to condense multiple instances of the same variable).

A polynomial is the sum (or difference) of monomials.

The degree of a monomial is the sum of the exponents of the variable symbols that appear in the monomial.

The degree of a polynomial is the degree of the monomial term with the highest degree.


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Name ___________________________________________________ Date____________________


Exit Ticket 1. Must the sum of three polynomials again be a polynomial?

2. Find (𝑤𝑤2 − 𝑤𝑤 + 1) + (𝑤𝑤3 − 2𝑤𝑤2 + 99).


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Exit Ticket Sample Solutions

1. Must the sum of three polynomials again be a polynomial?

Yes.

2. Find (𝒘𝒘𝟐𝟐 −𝒘𝒘 + 𝟏𝟏) + (𝒘𝒘𝟏𝟏 − 𝟐𝟐𝒘𝒘𝟐𝟐 + 𝟗𝟗𝟗𝟗).

𝒘𝒘𝟏𝟏 −𝒘𝒘𝟐𝟐 −𝒘𝒘 + 𝟏𝟏𝟏𝟏𝟏𝟏

Problem Set Sample Solutions

1. Celina says that each of the following expressions is actually a binomial in disguise:

i. 𝟓𝟓𝒂𝒂𝒂𝒂𝒂𝒂 − 𝟐𝟐𝒂𝒂𝟐𝟐 + 𝟏𝟏𝒂𝒂𝒂𝒂𝒂𝒂

ii. 𝟓𝟓𝒙𝒙𝟏𝟏 ∙ 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟏𝟏𝟏𝟏𝒙𝒙𝟒𝟒 + 𝟏𝟏𝒙𝒙𝟓𝟓 + 𝟏𝟏𝒙𝒙 ∙ (−𝟐𝟐)𝒙𝒙𝟒𝟒

iii. (𝒕𝒕+ 𝟐𝟐)𝟐𝟐 − 𝟒𝟒𝒕𝒕

iv. 𝟓𝟓(𝒂𝒂 − 𝟏𝟏) − 𝟏𝟏𝟏𝟏(𝒂𝒂 − 𝟏𝟏) + 𝟏𝟏𝟏𝟏𝟏𝟏(𝒂𝒂− 𝟏𝟏)

v. (𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐)𝟐𝟐 − (𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐) ∙ 𝟐𝟐𝟐𝟐

For example, she sees that the expression in (i) is algebraically equivalent to 𝟏𝟏𝟏𝟏𝒂𝒂𝒂𝒂𝒂𝒂 − 𝟐𝟐𝒂𝒂𝟐𝟐, which is indeed a binomial. (She is happy to write this as 𝟏𝟏𝟏𝟏𝒂𝒂𝒂𝒂𝒂𝒂+ (−𝟐𝟐)𝒂𝒂𝟐𝟐, if you prefer.)

Is she right about the remaining four expressions?

She is right about the remaining four expressions. They all can be expressed as binomials.

2. Janie writes a polynomial expression using only one variable, 𝒙𝒙, with degree 𝟏𝟏. Max writes a polynomial expression using only one variable, 𝒙𝒙, with degree 𝟕𝟕.

a. What can you determine about the degree of the sum of Janie’s and Max’s polynomials?

The degree would be 𝟕𝟕.

b. What can you determine about the degree of the difference of Janie’s and Max’s polynomials?

The degree would be 𝟕𝟕.

3. Suppose Janie writes a polynomial expression using only one variable, 𝒙𝒙, with degree of 𝟓𝟓, and Max writes a polynomial expression using only one variable, 𝒙𝒙, with degree of 𝟓𝟓.

a. What can you determine about the degree of the sum of Janie’s and Max’s polynomials?

The maximum degree could be 𝟓𝟓, but it could also be anything less than that. For example, if Janie’s polynomial were 𝒙𝒙𝟓𝟓 + 𝟏𝟏𝒙𝒙 − 𝟏𝟏, and Max’s were −𝒙𝒙𝟓𝟓 + 𝟐𝟐𝒙𝒙𝟐𝟐 + 𝟏𝟏, the degree of the sum is only 𝟐𝟐.

b. What can you determine about the degree of the difference of Janie’s and Max’s polynomials?

The maximum degree could be 𝟓𝟓, but it could also be anything less than that.


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4. Find each sum or difference by combining the parts that are alike.

a. (𝟐𝟐𝒑𝒑 + 𝟒𝟒) + 𝟓𝟓(𝒑𝒑 − 𝟏𝟏) − (𝒑𝒑+ 𝟕𝟕)

𝟏𝟏𝒑𝒑 − 𝟖𝟖

b. (𝟕𝟕𝒙𝒙𝟒𝟒 + 𝟗𝟗𝒙𝒙) − 𝟐𝟐(𝒙𝒙𝟒𝟒 + 𝟏𝟏𝟏𝟏)

𝟓𝟓𝒙𝒙𝟒𝟒 + 𝟗𝟗𝒙𝒙 − 𝟐𝟐𝟏𝟏

c. (𝟏𝟏 − 𝒕𝒕 − 𝒕𝒕𝟒𝟒) + (𝟗𝟗𝒕𝒕 + 𝒕𝒕𝟒𝟒)

𝟖𝟖𝒕𝒕+ 𝟏𝟏

d. (𝟓𝟓 − 𝒕𝒕𝟐𝟐) + 𝟏𝟏(𝒕𝒕𝟐𝟐 − 𝟖𝟖) − (𝒕𝒕𝟐𝟐 + 𝟏𝟏𝟐𝟐)

𝟒𝟒𝒕𝒕𝟐𝟐 − 𝟓𝟓𝟓𝟓

e. (𝟖𝟖𝒙𝒙𝟏𝟏 + 𝟓𝟓𝒙𝒙) − 𝟏𝟏(𝒙𝒙𝟏𝟏 + 𝟐𝟐)

𝟓𝟓𝒙𝒙𝟏𝟏 + 𝟓𝟓𝒙𝒙 − 𝟏𝟏

f. (𝟏𝟏𝟐𝟐𝒙𝒙 + 𝟏𝟏) + 𝟐𝟐(𝒙𝒙 − 𝟒𝟒) − (𝒙𝒙 − 𝟏𝟏𝟓𝟓)

𝟏𝟏𝟏𝟏𝒙𝒙 + 𝟖𝟖

g. (𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟓𝟓𝒙𝒙) − 𝟐𝟐(𝒙𝒙𝟐𝟐 + 𝟏𝟏)

𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟓𝟓𝒙𝒙 − 𝟐𝟐

h. (𝟗𝟗 − 𝒕𝒕 − 𝒕𝒕𝟐𝟐) − 𝟏𝟏𝟐𝟐 (𝟖𝟖𝒕𝒕+ 𝟐𝟐𝒕𝒕𝟐𝟐)

−𝟒𝟒𝒕𝒕𝟐𝟐 − 𝟏𝟏𝟏𝟏𝒕𝒕 + 𝟗𝟗

i. (𝟒𝟒𝒎𝒎+ 𝟏𝟏) − 𝟏𝟏𝟐𝟐(𝒎𝒎−𝟏𝟏) + (𝒎𝒎 + 𝟐𝟐)

−𝟕𝟕𝒎𝒎+ 𝟒𝟒𝟒𝟒

j. (𝟏𝟏𝟓𝟓𝒙𝒙𝟒𝟒 + 𝟏𝟏𝟏𝟏𝒙𝒙) − 𝟏𝟏𝟐𝟐(𝒙𝒙𝟒𝟒 + 𝟒𝟒𝒙𝒙)

𝟏𝟏𝒙𝒙𝟒𝟒 − 𝟏𝟏𝟖𝟖𝒙𝒙


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Lesson 9: Multiplying Polynomials

Student Outcomes

Students understand that the product of two polynomials produces another polynomial; students multiply polynomials.

Classwork


a. Gisella computed 𝟑𝟑𝟑𝟑𝟑𝟑× 𝟑𝟑𝟑𝟑 as follows:

Can you explain what she is doing? What is her final answer?

She is using an area model, finding the area of each rectangle and adding them together. Her final answer is 𝟕𝟕,𝟖𝟖𝟖𝟖𝟖𝟖.

Use a geometric diagram to compute the following products:

b. (𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙 + 𝟑𝟑) × (𝟑𝟑𝒙𝒙 + 𝟑𝟑)

𝟖𝟖𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟕𝟕𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟖𝟖𝒙𝒙 + 𝟖𝟖

c. (𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟏𝟏𝒙𝒙 + 𝟏𝟏)(𝒙𝒙𝟑𝟑 + 𝒙𝒙 + 𝟏𝟏)

𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟏𝟏𝒙𝒙 + 𝟏𝟏

d. (𝒙𝒙 − 𝟏𝟏)(𝒙𝒙𝟑𝟑 + 𝟖𝟖𝒙𝒙𝟑𝟑 − 𝟓𝟓)

𝒙𝒙𝟑𝟑 + 𝟓𝟓𝒙𝒙𝟑𝟑 − 𝟖𝟖𝒙𝒙𝟑𝟑 − 𝟓𝟓𝒙𝒙 + 𝟓𝟓

Ask the students:

What do you notice about the terms along the diagonals in the rectangles you drew?


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Encourage students to recognize that in parts (b) and (c), the terms along the diagonals were all like terms; however, in part (d) one of the factors has no 𝑥𝑥-term. Allow students to develop a strategy for dealing with this, concluding with the suggestion of inserting the term + 0𝑥𝑥, for a model that looks like the following:

Students may naturally ask about the division of polynomials. This topic will be covered in Grade 11, Module 1. The extension challenge at the end of the lesson, however, could be of interest to students inquiring about this.

Could we have found this product without the aid of a geometric model? What would that look like?

Go through the exercise applying the distributive property and collecting like terms. As a scaffold, remind students that variables are placeholders for numbers. If 𝑥𝑥 = 5, for example, whatever the quantity on the right is (270), you have 5 − 1 of “that quantity”, or 5 of “that quantity” minus 1 of “that quantity”. Similarly we have 𝑥𝑥 of that quantity, minus 1 of that quantity:

(𝑥𝑥 − 1)(𝑥𝑥3 + 6𝑥𝑥2 − 5)

𝑥𝑥(𝑥𝑥3 + 6𝑥𝑥2 − 5) − 1(𝑥𝑥3 + 6𝑥𝑥2 − 5)

𝑥𝑥4 + 6𝑥𝑥3 − 5𝑥𝑥 − 𝑥𝑥3 − 6𝑥𝑥2 + 5

𝑥𝑥4 + 5𝑥𝑥3 − 6𝑥𝑥2 − 5𝑥𝑥 + 5


Have students work Exercise 2 independently and then compare answers with a neighbor. If needed, facilitate agreement on the correct answer by allowing students to discuss their thought processes and justify their solutions.

Exercise 2

Multiply the polynomials using the distributive property: (𝟑𝟑𝒙𝒙𝟑𝟑 + 𝒙𝒙 − 𝟏𝟏)(𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙 + 𝟏𝟏).

𝟑𝟑𝒙𝒙𝟑𝟑(𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙 + 𝟏𝟏) + 𝒙𝒙(𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙 + 𝟏𝟏) − 𝟏𝟏(𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙 + 𝟏𝟏)

𝟑𝟑𝒙𝒙𝟖𝟖 − 𝟖𝟖𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟓𝟓 − 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝒙𝒙 − 𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙 − 𝟏𝟏

𝟑𝟑𝒙𝒙𝟖𝟖 + 𝒙𝒙𝟓𝟓 − 𝒙𝒙𝟑𝟑 − 𝟖𝟖𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙 − 𝟏𝟏


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Give students 10 minutes to complete Exercise 3 and compare their answers with a neighbor.

Exercise 3

The expression 𝟏𝟏𝟏𝟏𝒙𝒙𝟑𝟑 + 𝟖𝟖𝒙𝒙𝟑𝟑 is the result of applying the distributive property to the expression 𝟑𝟑𝒙𝒙𝟑𝟑(𝟓𝟓+ 𝟑𝟑𝒙𝒙). It is also the result of the applying the distributive property to 𝟑𝟑(𝟓𝟓𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙𝟑𝟑) or to 𝒙𝒙(𝟏𝟏𝟏𝟏𝒙𝒙 + 𝟖𝟖𝒙𝒙𝟑𝟑), for example, or even to 𝟏𝟏 ∙(𝟏𝟏𝟏𝟏𝒙𝒙𝟑𝟑 + 𝟖𝟖𝒙𝒙𝟑𝟑)!

For (i) to (x) below, write down an expression such that if you applied the distributive property to your expression it will give the result presented. Give interesting answers!

i. 𝟖𝟖𝒂𝒂 + 𝟏𝟏𝟑𝟑𝒂𝒂𝟑𝟑

ii. 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙𝟓𝟓 + 𝟑𝟑𝒙𝒙𝟏𝟏𝟏𝟏

iii. 𝟖𝟖𝒛𝒛𝟑𝟑 − 𝟏𝟏𝟓𝟓𝒛𝒛

iv. 𝟑𝟑𝟑𝟑𝒘𝒘𝟑𝟑 − 𝟏𝟏𝟑𝟑𝒘𝒘 + 𝟕𝟕𝟕𝟕𝒘𝒘𝟓𝟓

v. 𝒛𝒛𝟑𝟑(𝒂𝒂 + 𝒃𝒃) + 𝒛𝒛𝟑𝟑(𝒂𝒂+ 𝒃𝒃)

vi. 𝟑𝟑𝟑𝟑𝒔𝒔𝟑𝟑 +

𝟏𝟏𝟑𝟑

vii. 𝟏𝟏𝟓𝟓𝒑𝒑𝟑𝟑𝒓𝒓𝟑𝟑 − 𝟖𝟖𝒑𝒑𝟑𝟑𝒓𝒓𝟓𝟓 + 𝟗𝟗𝒑𝒑𝟑𝟑𝒓𝒓𝟑𝟑 + 𝟑𝟑√𝟑𝟑𝒑𝒑𝟑𝟑𝒓𝒓𝟖𝟖

viii. 𝟏𝟏.𝟑𝟑𝒙𝒙𝟗𝟗 − 𝟑𝟑𝟏𝟏𝒙𝒙𝟖𝟖

ix. (𝟑𝟑𝒙𝒙 + 𝟑𝟑)(𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟑𝟑) − (𝟑𝟑𝒙𝒙 + 𝟑𝟑)(𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟑𝟑)

x. (𝟑𝟑𝒛𝒛 + 𝟓𝟓)(𝒛𝒛 − 𝟑𝟑) − (𝟏𝟏𝟑𝟑𝒛𝒛 − 𝟑𝟑𝟖𝟖)(𝒛𝒛 − 𝟑𝟑)

Some possible answers:

i. 𝟑𝟑𝒂𝒂(𝟑𝟑+ 𝟕𝟕𝒂𝒂) or 𝟑𝟑(𝟑𝟑𝒂𝒂 + 𝟕𝟕𝒂𝒂𝟑𝟑) or 𝒂𝒂(𝟖𝟖+ 𝟏𝟏𝟑𝟑𝒂𝒂)

ii. 𝟑𝟑𝒙𝒙𝟑𝟑(𝟏𝟏+ 𝒙𝒙 + 𝒙𝒙𝟖𝟖) or 𝒙𝒙(𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙𝟗𝟗) or 𝟑𝟑(𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟓𝟓 + 𝒙𝒙𝟏𝟏𝟏𝟏)

iii. 𝟑𝟑𝒛𝒛(𝟑𝟑𝒛𝒛 − 𝟓𝟓) or 𝟑𝟑(𝟑𝟑𝒛𝒛𝟑𝟑 − 𝟓𝟓𝒛𝒛) or 𝒛𝒛(𝟖𝟖𝒛𝒛 − 𝟏𝟏𝟓𝟓)

iv. 𝟕𝟕𝒘𝒘(𝟖𝟖𝒘𝒘𝟑𝟑 − 𝟑𝟑 + 𝟏𝟏𝟏𝟏𝒘𝒘𝟑𝟑) or 𝒘𝒘(𝟑𝟑𝟑𝟑𝒘𝒘𝟑𝟑 − 𝟏𝟏𝟑𝟑 + 𝟕𝟕𝟕𝟕𝒘𝒘𝟑𝟑)

v. 𝒛𝒛𝟑𝟑�(𝒂𝒂+ 𝒃𝒃) + 𝒛𝒛(𝒂𝒂+ 𝒃𝒃)� or 𝒛𝒛(𝒛𝒛(𝒂𝒂 + 𝒃𝒃) + 𝒛𝒛𝟑𝟑(𝒂𝒂+ 𝒃𝒃))

vi. 𝟏𝟏𝟑𝟑

(𝟑𝟑𝒔𝒔𝟑𝟑 + 𝟏𝟏)

vii. 𝟑𝟑𝒑𝒑𝟑𝟑𝒓𝒓𝟑𝟑�𝟓𝟓𝒑𝒑𝒓𝒓𝟑𝟑 − 𝟑𝟑𝒓𝒓𝟑𝟑 + 𝟑𝟑𝒑𝒑𝟑𝟑 + √𝟑𝟑𝒑𝒑𝒓𝒓𝟑𝟑� or 𝒑𝒑𝟑𝟑𝒓𝒓𝟑𝟑(𝟏𝟏𝟓𝟓𝒑𝒑𝒓𝒓𝟑𝟑 − 𝟖𝟖𝒓𝒓𝟑𝟑 + 𝟗𝟗𝒑𝒑𝟑𝟑 + 𝟑𝟑√𝟑𝟑𝒑𝒑𝒓𝒓𝟑𝟑)

viii. 𝟏𝟏.𝟑𝟑𝒙𝒙𝟖𝟖(𝒙𝒙 − 𝟏𝟏𝟏𝟏𝟏𝟏) or 𝟑𝟑𝟏𝟏𝟏𝟏 𝒙𝒙𝟖𝟖(𝒙𝒙 − 𝟏𝟏𝟏𝟏𝟏𝟏)

ix. (𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟑𝟑)((𝟑𝟑𝒙𝒙+ 𝟑𝟑) − (𝟑𝟑𝒙𝒙 + 𝟑𝟑))

x. (𝒛𝒛 − 𝟑𝟑)�(𝟑𝟑𝒛𝒛+ 𝟓𝟓) − 𝟏𝟏𝟑𝟑(𝒛𝒛 − 𝟑𝟑)�

Choose one (or more) to go through as a class, listing as many different re-writes as possible. Then remark:

The process of making use of the distributive property “backwards” is factoring.


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Sammy wrote a polynomial using only one variable, 𝒙𝒙, of degree 𝟑𝟑. Myisha wrote a polynomial in the same variable of degree 𝟓𝟓. What can you say about the degree of the product of Sammy’s and Myisha’s polynomials?

The degree of the product of the two polynomials would be 𝟖𝟖.

Extension Extension

Find a polynomial that, when multiplied by 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙 + 𝟏𝟏, gives the answer 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙 − 𝟏𝟏.

𝒙𝒙 − 𝟏𝟏

Closing (6 minutes)

Is the product of two polynomials sure to be another polynomial?

Yes, by the definition of polynomial expression given in Lesson 8, the product of any two polynomial expressions is again a polynomial expression, which can then be written in standard polynomial form through application of the distributive property.

Is a polynomial squared sure to be another polynomial (other integer powers)? Yes.

Exit Ticket (4 minutes)


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Name ___________________________________________________ Date____________________


Exit Ticket 1. Must the product of three polynomials again be a polynomial?

2. Find (𝑤𝑤2 + 1)(𝑤𝑤3 − 𝑤𝑤 + 1).


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Exit Ticket Sample Solutions

1. Must the product of three polynomials again be a polynomial?

Yes.

2. Find (𝒘𝒘𝟑𝟑 + 𝟏𝟏)(𝒘𝒘𝟑𝟑 − 𝒘𝒘 + 𝟏𝟏).

𝒘𝒘𝟓𝟓 +𝒘𝒘𝟑𝟑 −𝒘𝒘 + 𝟏𝟏

Problem Set Sample Solutions

1. Use the distributive property to write each of the following expressions as the sum of monomials.

a. 𝟑𝟑𝒂𝒂(𝟑𝟑+ 𝒂𝒂)

𝟑𝟑𝒂𝒂𝟑𝟑 + 𝟏𝟏𝟑𝟑𝒂𝒂

b. 𝒙𝒙(𝒙𝒙+ 𝟑𝟑) + 𝟏𝟏

𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙 + 𝟏𝟏

c. 𝟏𝟏𝟑𝟑

(𝟏𝟏𝟑𝟑𝒛𝒛 + 𝟏𝟏𝟖𝟖𝒛𝒛𝟑𝟑)

𝟖𝟖𝒛𝒛𝟑𝟑 + 𝟑𝟑𝒛𝒛

d. 𝟑𝟑𝒙𝒙(𝒙𝒙𝟑𝟑 − 𝟏𝟏𝟏𝟏)

𝟑𝟑𝒙𝒙𝟑𝟑 − 𝟑𝟑𝟏𝟏𝒙𝒙

e. (𝒙𝒙 − 𝟑𝟑)(𝒙𝒙+ 𝟓𝟓)

𝒙𝒙𝟑𝟑 + 𝒙𝒙 − 𝟑𝟑𝟏𝟏

f. (𝟑𝟑𝒛𝒛 − 𝟏𝟏)(𝟑𝟑𝒛𝒛𝟑𝟑 + 𝟏𝟏)

𝟖𝟖𝒛𝒛𝟑𝟑 − 𝟑𝟑𝒛𝒛𝟑𝟑 + 𝟑𝟑𝒛𝒛 − 𝟏𝟏

g. (𝟏𝟏𝟏𝟏𝒘𝒘− 𝟏𝟏)(𝟏𝟏𝟏𝟏𝒘𝒘 + 𝟏𝟏)

𝟏𝟏𝟏𝟏𝟏𝟏𝒘𝒘𝟑𝟑 − 𝟏𝟏

h. (−𝟓𝟓𝒘𝒘− 𝟑𝟑)𝒘𝒘𝟑𝟑 −𝟓𝟓𝒘𝒘𝟑𝟑 − 𝟑𝟑𝒘𝒘𝟑𝟑

i. 𝟏𝟏𝟖𝟖𝒔𝒔𝟏𝟏𝟏𝟏𝟏𝟏 �𝟏𝟏𝟑𝟑 𝒔𝒔𝟑𝟑𝟏𝟏𝟏𝟏 + 𝟏𝟏.𝟏𝟏𝟑𝟑𝟓𝟓𝒔𝒔�

𝟖𝟖𝒔𝒔𝟑𝟑𝟏𝟏𝟏𝟏 + 𝟑𝟑𝒔𝒔𝟏𝟏𝟏𝟏𝟏𝟏

j. (𝟑𝟑𝒒𝒒 + 𝟏𝟏)(𝟑𝟑𝒒𝒒𝟑𝟑 + 𝟏𝟏)

𝟑𝟑𝒒𝒒𝟑𝟑 + 𝟑𝟑𝒒𝒒𝟑𝟑 + 𝟑𝟑𝒒𝒒 + 𝟏𝟏

k. (𝒙𝒙𝟑𝟑 − 𝒙𝒙 + 𝟏𝟏)(𝒙𝒙 − 𝟏𝟏)

𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟑𝟑𝒙𝒙 − 𝟏𝟏

l. 𝟑𝟑𝒙𝒙𝒛𝒛(𝟗𝟗𝒙𝒙𝒙𝒙+ 𝒛𝒛) − 𝟑𝟑𝒙𝒙𝒛𝒛(𝒙𝒙 + 𝒙𝒙 − 𝒛𝒛)

𝟑𝟑𝟕𝟕𝒙𝒙𝟑𝟑𝒙𝒙𝒛𝒛+ 𝟑𝟑𝒙𝒙𝒛𝒛𝟑𝟑 − 𝟑𝟑𝒙𝒙𝒙𝒙𝒛𝒛 − 𝟑𝟑𝒙𝒙𝟑𝟑𝒛𝒛+ 𝟑𝟑𝒙𝒙𝒛𝒛𝟑𝟑

m. (𝒕𝒕 − 𝟏𝟏)(𝒕𝒕+ 𝟏𝟏)(𝒕𝒕𝟑𝟑 + 𝟏𝟏)

𝒕𝒕𝟑𝟑 − 𝟏𝟏

n. (𝒘𝒘 + 𝟏𝟏)(𝒘𝒘𝟑𝟑 − 𝒘𝒘𝟑𝟑 +𝒘𝒘𝟑𝟑 −𝒘𝒘 + 𝟏𝟏)

𝒘𝒘𝟓𝟓 + 𝟏𝟏

o. 𝒛𝒛(𝟑𝟑𝒛𝒛+ 𝟏𝟏)(𝟑𝟑𝒛𝒛 − 𝟑𝟑)

𝟖𝟖𝒛𝒛𝟑𝟑 − 𝒛𝒛𝟑𝟑 − 𝟑𝟑𝒛𝒛

p. (𝒙𝒙 + 𝒙𝒙)(𝒙𝒙+ 𝒛𝒛)(𝒛𝒛+ 𝒙𝒙)

𝟑𝟑𝒙𝒙𝒙𝒙𝒛𝒛 + 𝒙𝒙𝟑𝟑𝒙𝒙+ 𝒙𝒙𝟑𝟑𝒛𝒛+ 𝒙𝒙𝒙𝒙𝟑𝟑 + 𝒙𝒙𝒛𝒛𝟑𝟑 + 𝒙𝒙𝟑𝟑𝒛𝒛 + 𝒙𝒙𝒛𝒛𝟑𝟑

q. 𝒙𝒙+𝒙𝒙𝟑𝟑

𝟏𝟏𝟑𝟑𝒙𝒙 +

𝟏𝟏𝟑𝟑𝒙𝒙

r. (𝟑𝟑𝟏𝟏𝒇𝒇𝟏𝟏𝟏𝟏 − 𝟏𝟏𝟏𝟏𝒇𝒇𝟓𝟓) ÷ 𝟓𝟓

𝟑𝟑𝒇𝒇𝟏𝟏𝟏𝟏 − 𝟑𝟑𝒇𝒇𝟓𝟓

s. −𝟓𝟓𝒙𝒙(𝒙𝒙𝟑𝟑 + 𝒙𝒙 − 𝟑𝟑) − 𝟑𝟑(𝟑𝟑 − 𝒙𝒙𝟑𝟑)

−𝟑𝟑𝒙𝒙𝟑𝟑 − 𝟓𝟓𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟏𝟏𝒙𝒙 − 𝟑𝟑 t.

(𝒂𝒂+𝒃𝒃−𝒄𝒄)(𝒂𝒂+𝒃𝒃+𝒄𝒄)𝟏𝟏𝟕𝟕

𝟏𝟏𝟏𝟏𝟕𝟕

𝒂𝒂𝟑𝟑 +𝟏𝟏𝟏𝟏𝟕𝟕

𝒃𝒃𝟑𝟑 −𝟏𝟏𝟏𝟏𝟕𝟕

𝒄𝒄𝟑𝟑 +𝟑𝟑𝟏𝟏𝟕𝟕

𝒂𝒂𝒃𝒃

u. (𝟑𝟑𝒙𝒙 ÷ 𝟗𝟗 + (𝟓𝟓𝒙𝒙) ÷ 𝟑𝟑) ÷ (−𝟑𝟑)

−𝟑𝟑𝟗𝟗𝒙𝒙𝟑𝟑𝟖𝟖

v. (−𝟑𝟑𝒇𝒇𝟑𝟑 − 𝟑𝟑𝒇𝒇 + 𝟏𝟏)(𝒇𝒇𝟑𝟑 − 𝒇𝒇 + 𝟑𝟑)

−𝟑𝟑𝒇𝒇𝟓𝟓 + 𝟑𝟑𝒇𝒇𝟑𝟑 − 𝟖𝟖𝒇𝒇𝟑𝟑 + 𝟑𝟑𝒇𝒇𝟑𝟑 − 𝟓𝟓𝒇𝒇 + 𝟑𝟑


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2. Use the distributive property (and your wits!) to write each of the following expressions as a sum of monomials. If the resulting polynomial is in one variable, write the polynomial in standard form.

a. (𝒂𝒂 + 𝒃𝒃)𝟑𝟑

𝒂𝒂𝟑𝟑 + 𝟑𝟑𝒂𝒂𝒃𝒃 + 𝒃𝒃𝟑𝟑

b. (𝒂𝒂 + 𝟏𝟏)𝟑𝟑

𝒂𝒂𝟑𝟑 + 𝟑𝟑𝒂𝒂 + 𝟏𝟏

c. (𝟑𝟑+ 𝒃𝒃)𝟑𝟑

𝒃𝒃𝟑𝟑 + 𝟖𝟖𝒃𝒃 + 𝟗𝟗

d. (𝟑𝟑+ 𝟏𝟏)𝟑𝟑

𝟏𝟏𝟖𝟖

e. (𝒙𝒙 + 𝒙𝒙+ 𝒛𝒛)𝟑𝟑

𝒙𝒙𝟑𝟑 + 𝒙𝒙𝟑𝟑 + 𝒛𝒛𝟑𝟑 + 𝟑𝟑𝒙𝒙𝒙𝒙+ 𝟑𝟑𝒙𝒙𝒛𝒛 + 𝟑𝟑𝒙𝒙𝒛𝒛

f. (𝒙𝒙 + 𝟏𝟏+ 𝒛𝒛)𝟑𝟑

𝒙𝒙𝟑𝟑 + 𝒛𝒛𝟑𝟑 + 𝟑𝟑𝒙𝒙𝒛𝒛+ 𝟑𝟑𝒙𝒙 + 𝟑𝟑𝒛𝒛 + 𝟏𝟏

g. (𝟑𝟑+ 𝒛𝒛)𝟑𝟑

𝒛𝒛𝟑𝟑 + 𝟖𝟖𝒛𝒛 + 𝟗𝟗

h. (𝒑𝒑 + 𝒒𝒒)𝟑𝟑

𝒑𝒑𝟑𝟑 + 𝟑𝟑𝒑𝒑𝟑𝟑𝒒𝒒 + 𝟑𝟑𝒑𝒑𝒒𝒒𝟑𝟑 + 𝒒𝒒𝟑𝟑

i. (𝒑𝒑 − 𝟏𝟏)𝟑𝟑

𝒑𝒑𝟑𝟑 − 𝟑𝟑𝒑𝒑𝟑𝟑 + 𝟑𝟑𝒑𝒑 − 𝟏𝟏

j. (𝟓𝟓+ 𝒒𝒒)𝟑𝟑

𝒒𝒒𝟑𝟑 + 𝟏𝟏𝟓𝟓𝒒𝒒𝟑𝟑 + 𝟕𝟕𝟓𝟓𝒒𝒒+ 𝟏𝟏𝟑𝟑𝟓𝟓

3. Use the distributive property (and your wits!) to write each of the following expressions as a polynomial in standard form.

a. (𝒔𝒔𝟑𝟑 + 𝟑𝟑)(𝒔𝒔 − 𝟏𝟏)

𝒔𝒔𝟑𝟑 − 𝒔𝒔𝟑𝟑 + 𝟑𝟑𝒔𝒔 − 𝟑𝟑

b. 𝟑𝟑(𝒔𝒔𝟑𝟑 + 𝟑𝟑)(𝒔𝒔 − 𝟏𝟏)

𝟑𝟑𝒔𝒔𝟑𝟑 − 𝟑𝟑𝒔𝒔𝟑𝟑 + 𝟏𝟏𝟑𝟑𝒔𝒔 − 𝟏𝟏𝟑𝟑

c. 𝒔𝒔(𝒔𝒔𝟑𝟑 + 𝟑𝟑)(𝒔𝒔 − 𝟏𝟏)

𝒔𝒔𝟑𝟑 − 𝒔𝒔𝟑𝟑 + 𝟑𝟑𝒔𝒔𝟑𝟑 − 𝟑𝟑𝒔𝒔

d. (𝒔𝒔 + 𝟏𝟏)(𝒔𝒔𝟑𝟑 + 𝟑𝟑)(𝒔𝒔 − 𝟏𝟏)

𝒔𝒔𝟑𝟑 + 𝟑𝟑𝒔𝒔𝟑𝟑 − 𝟑𝟑

e. (𝒖𝒖 − 𝟏𝟏)(𝒖𝒖𝟓𝟓 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖 + 𝟏𝟏)

𝒖𝒖𝟖𝟖 − 𝟏𝟏 f. √𝟓𝟓(𝒖𝒖− 𝟏𝟏)(𝒖𝒖𝟓𝟓 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖+ 𝟏𝟏)

√𝟓𝟓𝒖𝒖𝟖𝟖 − √𝟓𝟓

g. (𝒖𝒖𝟕𝟕 + 𝒖𝒖𝟑𝟑 + 𝟏𝟏)(𝒖𝒖− 𝟏𝟏)(𝒖𝒖𝟓𝟓 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖𝟑𝟑 + 𝒖𝒖 + 𝟏𝟏)

𝒖𝒖𝟏𝟏𝟑𝟑 + 𝒖𝒖𝟗𝟗 − 𝒖𝒖𝟕𝟕 + 𝒖𝒖𝟖𝟖 − 𝒖𝒖𝟑𝟑 − 𝟏𝟏

4. Beatrice writes down every expression that appears in this problem set, one after the other, linking them with “+” signs between them. She is left with one very large expression on her page. Is that expression a polynomial expression? That is, is it algebraically equivalent to a polynomial?

Yes.

What if she wrote “−” signs between the expressions instead?

Yes.

What if she wrote “×” signs between the expressions instead?

Yes.


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M1 Mid-Module Assessment Task

ALGEBRA I

8. a. Find the following products:

i. (𝑥𝑥 − 1)(𝑥𝑥 + 1)

ii. (𝑥𝑥 − 1)�𝑥𝑥2 + 𝑥𝑥 + 1� iii. (𝑥𝑥 − 1)(𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) iv. (𝑥𝑥 − 1)(𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) v. (𝑥𝑥 − 1)(𝑥𝑥𝑛𝑛 + 𝑥𝑥𝑛𝑛−1 +⋯+ 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1)

b. Substitute 𝑥𝑥 = 10 into each of the products from parts (i) through (iv) and your answers to show

how each of the products appears as a statement in arithmetic.

c. If we substituted 𝑥𝑥 = 10 into the product (𝑥𝑥 − 2)(𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) and computed the product, what number would result?

Module 1: Relationships Between Quantities and Reasoning with Equations and Their Graphs

108







ALGEBRA I

d. Multiply (𝑥𝑥 − 2) and (𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1), and express your answer in standard form. Substitute 𝑥𝑥 = 10 into your answer, and see if you obtain the same result that you obtained in part (c).

e. Francois says (𝑥𝑥 − 9)(𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) must equal 𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 +𝑥𝑥2 + 𝑥𝑥 + 1 because when 𝑥𝑥 = 10, multiplying by “𝑥𝑥 − 9” is the same as multiplying by 1.

i. Multiply (𝑥𝑥 − 9)(𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1). ii. Put 𝑥𝑥 = 10 into your answer.

Is it the same as 𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1 with 𝑥𝑥 = 10? iii. Was Francois right?


109







ALGEBRA I

8. a. Find the following products:

i. (𝑥𝑥 − 1)(𝑥𝑥 + 1)

x2 + x − x− 1 x2-1

ii. (𝑥𝑥 − 1)�𝑥𝑥2 + 𝑥𝑥 + 1� x3 + x2 + x − x2 − x− 1 x3 − 1

iii. (𝑥𝑥 − 1)(𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) x4 + x3 + x2 + x − x3 − x2 − x −1 x4 − 1

iv. (𝑥𝑥 − 1)(𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) x5 + x4 + x3 + x2 + x − x4 − x3 − x2 − x − 1 x5 − 1

v. (𝑥𝑥 − 1)(𝑥𝑥𝑛𝑛 + 𝑥𝑥𝑛𝑛−1 +⋯+ 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) xn+1 − 1

b. Substitute 𝑥𝑥 = 10 into each of the products from parts (i) through (iv) and your answers to show

how each of the products appears as a statement in arithmetic.

i. (10 − 1) ∙ (10 + 1) = (100 − 1) 9 ∙ (11) = 99

ii. (10 − 1) ∙ (100 + 10 + 1) = (1000− 1) 9 ∙ (111) = 999

iii. (10 − 1) ∙ (1,000 + 100 + 10 + 1) = (10,000 − 1) 9 ∙ (1,111) = 9,999

iv. (10 − 1) ∙ (10,000 + 1,000 + 100 + 10 + 1) = (100,000 − 1) 9 ∙ (11,111) = 99,999

c. If we substituted 𝑥𝑥 = 10 into the product (𝑥𝑥 − 2)(𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) and

computed the product, what number would result?

8 ∙ (11,111,111) = 88,888,888


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brittjm

Sticky Note

Answer key


ALGEBRA I

d. Multiply (𝑥𝑥 − 2) and (𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) and express your answer in standard form.

x8 + x7 + x6 + x5 + x4 + x3 + x2 + x − 2x7 − 2x6 − 2x5 − 2x4 −2x3 −2x2 − 2x− 2 x8 − x7 − x6 − x5 − x4 − x3 − x2 − x −2

Substitute 𝑥𝑥 = 10 into your answer and see if you obtain the same result as you obtained in part (c).

108 −107 −106 −105 − 104 − 103 − 102 − 10− 2 = 88,888,888. Yes, I get the same answer.

e. Francois says (𝑥𝑥 − 9)(𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) must equal 𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 +𝑥𝑥2 + 𝑥𝑥 + 1 because when 𝑥𝑥 = 10, multiplying by "𝑥𝑥 − 9" is the same as multiplying by 1.

i. Multiply (𝑥𝑥 − 9)(𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1).

x8 − 8x7 − 8x6 − 8x5 − 8x4 −8x3 −8x2 − 8x− 9

ii. Put 𝑥𝑥 = 10 into your answer.

100,000,000− 80,000,000− 8,000,000− 800,000− 80,000 −8,000 − 800− 80− 9

100,000,000− 88,888,889 = 11,111,111

Is it the same as 𝑥𝑥7 + 𝑥𝑥6 + 𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1 with 𝑥𝑥 = 10?

Yes.

iii. Was Francois right?

No, just because it is true when x is 10, doesn’t make it true for all real x. The two

expressions are not algebraically equivalent.


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lesson 8: adding and subtracting polynomials

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