MAFS Algebra 1

Algebraic Expressions & Polynomials

Day 4 - Student Packet

Adapted from: 1

The Distributive Property: If a, b, and c are real numbers, then 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐.

The distributive property represents a key belief about the arithmetic of real numbers. This property can be applied to

algebraic expressions using variables that represent real numbers.

Day 4: Algebraic Expressions & Polynomials MAFS.912.A-SSE.1.1, MAFS.912.A-APR.1.1 I CAN …

rewrite algebraic expressions in different equivalent forms by recognizing the expression’s structure

simplify expressions (e.g., combining like terms, using the distributive property, and other operations with polynomials)

add, subtract , and multiply polynomials with integral coefficients

apply the understanding of closure to adding, subtracting, and multiplying polynomials with integral coefficients

Algebraic Expressions—The Distributive Property

Problem Set

1. Insert parentheses to make each statement true.

a. + + =

b. + + =

c. + + =

d. + + =

2. Using starting symbols of , , , and , which of the following expressions will NOT appear when following the rules of this

game?

Luke wants to play the 4-number game with the numbers 1, 2, 3, and 4 and the operations of addition, multiplication, AND

subtraction.

Leoni responds, “Or we just could play the 4-number game with just the operations of addition and multiplication, but now

with the numbers -1, -2, -3, -4, 1, 2, 3, and 4 instead.”

a. + + ( )

b.

c.

d. +

e. +

3. Consider the expression: ( + ) ( + ) ( + ).

a. Draw a picture to represent the expression.

b. Write an equivalent expression by applying the distributive property.

Adapted from: 2

4. Given that , which of the shaded regions is larger and why?

5. Consider the following diagram.

Edna looked at the diagram and then highlighted the four small rectangles shown and concluded:

( + ) = + ( + ).

a. Michael, when he saw the picture, highlighted four rectangles and concluded:

( + ) = + + ( + ).

Which four rectangles and one square did he highlight?

b. Jill, when she saw the picture, highlighted some rectangles and squares to conclude:

( + ) = + + .

Which rectangles and squares did she highlight?

c. When Fatima saw the picture, she exclaimed: ( + ) = + ( + ) . She claims she highlighted just four

rectangles to conclude this. Identify the four rectangles she highlighted and explain how using them she arrived that the

expression + ( + ) .

d. Is each student's technique correct? Explain why or why not.

Adapted from: 3

The Commutative Property of Addition: If 𝑎 and 𝑏 are real numbers, then 𝑎 + 𝑏 = 𝑏 + 𝑎.

The Associative Property of Addition: If 𝑎, 𝑏, and 𝑐 are real numbers, then (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐).

The Commutative Property of Multiplication: If 𝑎 and 𝑏 are real numbers, then 𝑎 𝑏 = 𝑏 𝑎.

The Associative Property of Multiplication: If 𝑎, 𝑏, and 𝑐 are real numbers, then (𝑎𝑏)𝑐 = 𝑎(𝑏𝑐).

The commutative and associative properties represent key beliefs about the arithmetic of real numbers. These properties

can be applied to algebraic expressions using variables that represent real numbers.

Numerical Symbol: A numerical symbol is a symbol that represents a specific number.

Variable Symbol: A variable symbol is a symbol that is a placeholder for a number.

Algebraic Expression: An algebraic expression is either

1. A numerical symbol or a variable symbol, or

2. The result of placing previously generated algebraic expressions into the two blanks of one of the four operators

((__)+(__), (__)-(__), (__)×(__),(__)÷(__)) or into the base blank of an exponentiation with exponent that is a

rational number.

Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the

commutative, associative, and distributive properties and the properties of rational exponents to components of the first

expression.

Algebraic Expressions—The Commutative and Associative Properties

6. Fill in the blanks of this proof showing that ( + )( + ) is equivalent + + . Write either “commutative property,”

“associative property,” or “distributive property” in each blank.

( + )( + ) = ( + ) + ( + )

= ( + ) + ( + )

= ( + ) + ( + )

= + + ( + )

= + + ( + )

= + + +

= + ( + ) +

= + +

7. What is a quick way to see that the value of the sum + + + is ?

8.

a. If = and =

, what is the value of the product ?

b. Give some indication as to how you used the commutative and associative properties of multiplication to evaluate

in part (a).

c. Did you use the associative and commutative properties of addition to answer Question 7?

Adapted from: 4

9. The following is a proof of the algebraic equivalency of ( ) and . Fill in each of the blanks with either the statement

“commutative property” or “associative property.”

( ) =

= ( )( )

= ( )( )

= ( )

= ( )

= ( )( )

=

Interpreting Algebraic Expressions

10. Match the expressions, words, tables, and areas

Expressions Words Tables Areas

+

Multiply n by two, then add six.

n 1 2 3 4

Ans 14 16 18 20

Multiply n by three, then square the answer.

n 1 2 3 4

Ans 81 144

+ Add six to n then multiply by two.

n 1 2 3 4

Ans 10 15 22

+ Add six to n then divide by wo.

n 1 2 3 4

Ans 3 27 48

( + ) Add three to n then multiply by two.

n 1 2 3 4

Ans 81 10

+

Add six to n then square the answer.

n 1 2 3 4

Ans 10 12 14

( ) Multiply n by two then add twelve.

n 1 2 3 4

Ans 4 5

( + ) Divide n by two then add six. n 1 2 3 4

Ans 6.5 7 7.5 8

+ + Square n, then add six

+

+

+

Adapted from: 5

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.

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.

A polynomial expression with one variable symbol 𝑥 is in standard form if it is expressed as, 𝑎𝑛𝑥𝑛 + 𝑎𝑛− 𝑥

𝑛− +⋯+ 𝑎 𝑥 +

𝑎0, where 𝑛 is a non-negative integer, and 𝑎0, 𝑎 , 𝑎 , … , 𝑎𝑛 are constant coefficients with 𝑎𝑛 ≠ . A polynomial expression in

𝑥 that is in standard form is often called a polynomial in 𝑥.

A set is closed under an operation if the operation on any two elements of the set produces another element of the set. If an

element outside the set is produced, then the set is not closed under the operation.

Polynomials will be closed under an operation if the operation produces another polynomial.

When adding or subtracting polynomials, the variables and their exponents do not change. Only their coefficients will possibly

change. This guarantees that the sum or difference has variables and exponents which are already classified as belonging to

polynomials. Polynomials are closed under addition and subtraction.

Polynomials

Adding and Subtracting Polynomials

Problem Set

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

i. +

ii. + + ( )

iii. ( + )

iv. ( ) ( ) + ( )

v. ( ) ( )

For example, she sees that the expression in (i) is algebraically equivalent to , which is indeed a binomial. (She is

happy to write this as + ( ) , if you prefer.)

Is she right about the remaining four expressions?

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

one variable, , with degree .

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

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

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

expression using only one variable, , with degree of .

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

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

Adapted from: 6

When multiplying polynomials, the variables' exponents are added, according to the rules of exponents. Remember that the

exponents in polynomials are whole numbers. The whole numbers are closed under addition, which guarantees that the new

exponents will be whole numbers. Consequently, polynomials are closed under multiplication.

When dividing variables with exponents, there is the possibility of creating a negative exponent. Negative exponents are not

allowed in polynomials. Consequently, polynomials are NOT closed under division.

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

a. ( + ) + ( ) ( + )

b. ( + ) ( + )

c. ( ) + ( + )

d. ( ) + ( ) ( + )

e. ( + ) ( + )

f. ( + ) + ( ) ( )

g. ( + ) ( + )

h. ( )

( + )

i. ( + ) ( ) + ( + )

j. ( + ) ( + )

Multiplying Polynomials

Problem Set

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

a. ( + )

b. ( + ) +

c.

( + )

d. ( )

e. ( )( + )

f. ( )( + )

g. ( )( + )

h. ( )

i. ( + )( + )

j. ( + ) ( + )

k. ( + )( + + )

l. ( + )( )

m.

n. ( 0 )

o. ( + ) ( )

p. ( − )( )

6. 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. ( + )

b. ( + )

c. ( + )

d. ( + )

e. ( + + )

f. ( + )

g. ( + + )

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

a. ( + )( )

b. ( + )( )

c. ( + )( )

d. ( + )( + )( )

e. ( )( + + + + + )

f. √ ( )( + + + + + )

g. ( + + )( )( + + + + + )

Adapted from: 7

8. 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?

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

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

9. Lia is creating a banner for her brother’s surprise birthday party. The banner will be a large rectangle, divided into sections for

text and pictures. The figure below represents Lia’s design for the banner. Use the diagram to answer the questions that follow.

All units are in inches.

a. What is the perimeter of the shaded portion of the banner?

b. What is the area of the entire rectangular banner?