Algebra 1 Workbook — Chapter 12: Adding, Subtracting, and Multiplying Polynomials

SOL A.EO.2 (a, b) · Companion to Textbook Chapter 12

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 130.

PAGE 1 — Chapter opener

Chapter 12 · Adding, Subtracting, and Multiplying Polynomials

Standard A.EO.2 (a, b)

In this chapter you will:

Name terms, coefficients, degree, leading coefficient, and constant term, and write a polynomial in standard form
Classify a polynomial as a monomial, binomial, or trinomial, and by its degree
Combine like terms — and say which terms are allowed to combine
Add polynomials with algebra tiles, a vertical arrangement, and symbols
Subtract by adding the opposite, using zero pairs on a tile mat
Multiply a monomial by a polynomial with the distributive property and an area model
Multiply two polynomials with an area model, a tile rectangle, and symbols (factors of five or fewer terms)
Move in both directions between tiles, area models, and symbols
Check any answer by evaluating the original and your simplified answer at the same input

Words to know: polynomial · term · coefficient · degree · standard form · leading coefficient · constant term · monomial · binomial · trinomial · like terms · algebra tile · zero pair · opposite of a polynomial · distributive property · area model

Convention: the sign in front belongs to the term. In $5x^{3} - 2x^{2} + x - 8$ the four terms are $5x^{3}$ , $-2x^{2}$ , $x$ , and $-8$ .

Convention: only like terms combine, and combining them never changes the exponent: $3x^{2} + 5x^{2} = 8x^{2}$ , not $8x^{4}$ .

Convention: subtracting is adding the opposite — every sign of the second polynomial changes. Multiplying uses $a^{m} \cdot a^{n} = a^{m+n}$ from Chapter 10.

Calculator. Algebra 1 has no no-calculator standards. Use one to confirm a check: pick $x = 2$ and $x = -1$ , evaluate the original and your answer, and see whether the two numbers agree.

PAGE 2 — Anatomy of a polynomial

12.1 Polynomials, Terms, and Standard Form

FIGURE: fig1-anatomy-of-a-polynomial.png (full width)

Fill in the blanks from the figure.

A term is a piece joined by $+$ or $-$ , and the ____________ in front belongs to it.

The degree of a polynomial is the ____________ exponent of any of its terms.

Standard form writes the terms with exponents ____________.

Name the four terms of $5x^{3} - 2x^{2} + x - 8$ and give the coefficient of each.

Terms: ____________ ____________ ____________ ____________

Coefficients: ______ ______ ______ ______
Degree: ______ Leading coefficient: ______ Constant term: ______
What does it mean to say this polynomial is written in standard form, and where do you see it?
From the table in the figure, name a monomial, a binomial, and a trinomial, and give the degree of each.

Monomial: ____________ deg ______ Binomial: ____________ deg ______

Trinomial: ____________ deg ______

PAGE 3 — Like terms

Sorting Like Terms

FIGURE: fig2-like-terms-sorting.png (full width)

Complete the frame. Like terms have the same variable raised to the ____________ exponent. Only the ____________ are added; the exponent ____________ the pile.

Which three terms went into the $x^{2}$ bin, and what did they combine to?

____________ + ____________ + ____________ = ____________
Why were $-3x$ and $4x^{2}$ not put in the same bin?
Combine $4x^{2} - 3x + 7 + x^{2} + 5x - 2 - 6x^{2}$ in standard form: ____________
Explain. Why can $5x^{2}$ and $5x^{3}$ not be combined, even though they share a coefficient and a variable? Refer to what the exponent is counting.

PAGE 4 — Practice · naming and standard form

Practice · Parts and Standard Form

Give the degree, the leading coefficient, and the constant term of each.

Polynomial	Degree	Leading coeff.	Constant
a) $7x^{2} - 3x + 1$
b) $-4x^{3} + x$
c) $9$
d) $x^{4} - 6x^{5} + 2$

Write each in standard form and say how many terms it has.

Expression	Standard form	# of terms
a) $4x - 9 + x^{2}$
b) $5 - 2x^{3}$
c) $x + 7x^{3} - 2x^{2} + 6$
d) $-11x^{2}$

Classify each by its number of terms, then by its degree.

Polynomial	By terms	By degree
a) $3x^{2}$
b) $x^{2} - 4$
c) $2x^{2} + 5x - 3$
d) $x^{3} + x^{2} + x + 1$

PAGE 5 — Practice · combining and evaluating

Practice · Combine and Evaluate

Combine like terms.

Expression	Simplified
a) $6x + 2x$
b) $9x^{2} - 4x^{2}$
c) $3x^{2} + 5x - x^{2} + 2x$
d) $8 - 3x + 4x - 15$

Evaluate $P(x) = 2x^{2} - 5x + 3$ at $x = 2$ : ______ at $x = -1$ : ______
Apply it. $h(t) = -16t^{2} + 40t + 5$ . Degree: ______ Leading coeff.: ______ Constant: ______

What the constant term means about the throw: _______________________________________
Which is not a polynomial in one variable, and why?

a) $x^{2} - 7$ b) $3x^{-2} + 1$ c) $\tfrac12 x^{3} - x$ d) $4$

Not a polynomial: ______ Why: _______________________________________
Find the error. A student says the degree of $6x + x^{4} - 2$ is $1$ , "because the first term is $6x$ ."

What went wrong? _______________________________________

Standard form: ____________ Correct degree: ______
Coefficient of $x$ in each: a) $x^{2} - x + 4$ ______ b) $3x^{2} + 7$ ______

c) $\tfrac12 x - 9$ ______ d) $-x^{2} - 8x$ ______
Apply it. $R(x) = -2x^{2} + 90x$ . Degree: ______ LC: ______ Constant: ______

What the constant says about charging nothing: _______________________________________

PAGE 6 — Exit ticket 12.1

Exit Ticket · Lesson 12.1

Name: ________________________ Date: ____________

Write $3 - x^{2} + 5x^{3}$ in standard form: ____________

Degree: ______ Leading coefficient: ______
Combine: $7x^{2} - 2x + 4 - 3x^{2} + 9x =$ ____________
Classify $4x^{2} - 9$ by terms: ____________ by degree: ____________
Evaluate $x^{3} - 4x + 1$ at $x = -2$ : ______

PAGE 7 — Adding with tiles

12.2 Adding Polynomials

FIGURE: fig3-adding-with-tiles.png (full width)

Fill in the blanks from the figure.

A large square is worth ______, a long tile is worth ______, and a small square is worth ______.

To add, push the collections together and count each ____________.

Name the polynomial each of the two upper tile collections represents.

First: ____________ Second: ____________
How many of each shape are in the combined collection, and what polynomial is that?

Large: ______ Long: ______ Small: ______ Polynomial: ____________
Why are the long tiles never counted together with the small squares?
Describe the tiles for $2x^{2} + x + 2$ : ______ large, ______ long, ______ small.
A collection has $1$ large square, $4$ long tiles, and $3$ small squares. Polynomial: ____________

PAGE 8 — Practice · adding symbolically

Practice · Drop the Parentheses

$(3x^{2} + 2x) + (x^{2} + 5x) =$ ____________
Add.

Sum	Answer
a) $(2x + 5) + (3x - 1)$
b) $(x^{2} + 4x - 2) + (3x^{2} - x + 6)$
c) $(5x^{3} - x) + (2x^{3} + 7x - 4)$
d) $(-3x^{2} + 8) + (3x^{2} - 8)$

Add $(4x^{3} - 2x^{2} + 9) + (x^{3} + 6x^{2} - 5x)$ vertically. Show the columns.

	$x^{3}$	$x^{2}$	$x$	const
first
second
sum

Answer: ____________

$\left(\tfrac12 x^{2} + 3x\right) + \left(\tfrac32 x^{2} - x\right) =$ ____________
$(x^{2} + 1) + (2x^{2} - 3x) + (x - 4) =$ ____________

PAGE 9 — Practice · tiles, context, and checks

Practice · Both Directions and Context

Tile picture for $(x^{2} + 3x + 1) + (2x^{2} + x + 2)$ :

First: ______ large, ______ long, ______ small

Second: ______ large, ______ long, ______ small

Total: ______ large, ______ long, ______ small Sum: ____________
Collections: $2$ large, $5$ long, $4$ small and $1$ large, $2$ long, $1$ small.

Polynomials: ____________ and ____________ Sum: ____________
Apply it. $P_A(x) = 2x^{2} + 30x - 50$ , $P_B(x) = x^{2} - 10x + 120$ . Combined profit: ____________
Apply it. Sides $x^{2} + 3$ , $2x + 1$ , and $x^{2} + x - 4$ . Perimeter: ____________
Explain. How can the sum of two degree- $2$ polynomials have degree less than $2$ ? Give an example.
Find the error. A student writes $(3x^{2} + 2x) + (x^{2} + 5x) = 4x^{4} + 7x^{2}$ .

What went wrong? _______________________________________ Correct: ____________
Check 29b at $x = 2$ . Original: ______ Answer: ______
$(2x^{2} - x + 5) + \underline{\hspace{2.5cm}} = 5x^{2} + 3x + 1$

PAGE 10 — Exit ticket 12.2

Exit Ticket · Lesson 12.2

Name: ________________________ Date: ____________

$(6x^{2} - 4x + 1) + (2x^{2} + 9x - 7) =$ ____________
$3$ large, $1$ long, $5$ small. Polynomial: ____________
Tile picture for $(x^{2} + 2x) + (x^{2} + 3)$ and the sum:

_______________________________________________ Sum: ____________
Check 41 at $x = -1$ . Original: ______ Answer: ______

PAGE 11 — Zero pairs

12.3 Subtracting Polynomials

FIGURE: fig4-zero-pairs-cancel.png (full width)

Complete the frame. A white tile is the ____________ of the shaded tile of the same shape. A shaded tile beside its white partner is a ____________ and is worth ______.

What is a zero pair, and why may one always be removed from a collection?
In the right-hand panel, explain why $3x + (-x) = 2x$ by describing the tiles.

PAGE 12 — Subtracting with tiles

Adding the Opposite on the Mat

FIGURE: fig5-subtracting-with-tiles.png (full width)

The four moves: lay out the first polynomial → add the ____________ of the second → remove every ____________ → read what is left.

What is the opposite of $x^{2} + 4x + 1$ , and how does the figure show it?

Opposite: ____________
Name every zero pair that was removed, and name the tiles that were left.

Zero pairs: _______________________________________

Left: _______________________________________ Polynomial: ____________
Carry out $(2x^{2} + x + 3) - (x^{2} + 4x + 1)$ symbolically: ____________

PAGE 13 — Distribute the minus sign

Every Sign Changes

FIGURE: fig6-distribute-the-minus-sign.png (full width)

Remember. The minus sign in front of parentheses is a $-1$ waiting to be ____________, and it reaches ____________ term inside.

What exactly did the student who got $2x^{2} + 3x - 3$ do wrong?
How does the row of the table at $x = 2$ show that $2x^{2} - 9x + 13$ is right and $2x^{2} + 3x - 3$ is not?
Write the opposite of each.

Polynomial	Opposite
a) $3x - 7$
b) $-x^{2} + 5x - 2$
c) $4x^{2}$
d) $-6$

PAGE 14 — Practice · subtracting

Practice · Change Every Sign

Subtract.

Difference	Answer
a) $(7x + 3) - (2x + 8)$
b) $(5x^{2} - x + 4) - (2x^{2} + 3x - 6)$
c) $(x^{3} + 2x) - (x^{3} - 5x + 1)$
d) $(4x^{2} - 9) - (4x^{2} - 9)$

Rewrite $(6x^{3} - 2x + 5) - (x^{3} + 4x^{2} - 2x)$ as an addition, then subtract vertically.

As addition: _______________________________________

Answer: ____________
$(x^{2} - 6x) - (3x^{2} + 2x - 8) =$ ____________
Apply it. $R(x) = -2x^{2} + 90x$ , $C(x) = 20x + 150$ . Profit $R - C$ : ____________
Apply it. Pipe $5x + 12$ inches; piece $2x - 3$ cut away. Remaining length: ____________

PAGE 15 — Practice · reasoning and checks

Practice · Why Every Sign Matters

Explain. Why is subtracting a multi-term polynomial not the same as subtracting only its first term? Use $(x^{2} + 1) - (x^{2} - 4)$ .

Correct answer: ______ Careless answer: ______
Find the error. A student computes $(8x^{2} - 3x) - (5x^{2} - 7x) = 3x^{2} - 10x$ .

What went wrong? _______________________________________ Correct: ____________
Check 52b at $x = 2$ and at $x = -1$ .

Input	Original	Answer
$x = 2$
$x = -1$

$(5x^{2} + 2x - 1) - \underline{\hspace{2.5cm}} = 2x^{2} + 6x$
Tile picture for $(3x^{2} + 2x + 1) - (x^{2} + 3x)$ :

Starting tiles: _______________________________________

Opposite tiles added: _______________________________________

Zero pairs removed: _______________________________________

Left: _______________________________________ Difference: ____________

PAGE 16 — Exit ticket 12.3

Exit Ticket · Lesson 12.3

Name: ________________________ Date: ____________

$(9x^{2} - 5x + 2) - (4x^{2} + x - 6) =$ ____________
Opposite of $-2x^{2} + 7x - 3$ : ____________
Check 63 at $x = 2$ . Original: ______ Answer: ______
Find the error. A student says $(x^{2} - 4) - (x^{2} + 4) = 0$ , "because the $x^{2}$ terms cancel and so do the fours."

What went wrong? _______________________________________ Correct: ____________

PAGE 17 — Monomial area model

12.4 Multiplying by a Monomial

FIGURE: fig7-area-model-monomial-times-trinomial.png (full width)

Complete the frame. Multiply the ____________; add the ____________. A one-row area model never produces ____________ terms — the answer is the list of cells.

Name the row label, the three column labels, and the three cells.

Row: ______ Columns: ______ ______ ______

Cells: ______ ______ ______
Which law of exponents turned $2x$ and $3x^{2}$ into $6x^{3}$ , and what does that law say?

Law: ____________ Statement: ____________
Why can the three cells not be combined into fewer terms?
$3x(x + 5) =$ ____________
Area model for $4(2x^{2} - x)$ — row, columns, cells, product:

Row: ______ Columns: ______ ______

Cells: ______ ______ Product: ____________
$-2x(3x - 7) =$ ____________

PAGE 18 — Practice · distribute

Practice · Every Term Gets Multiplied

Multiply.

Product	Answer
a) $5(2x - 3)$
b) $x(x^{2} + 4)$
c) $3x(2x^{2} - x + 6)$
d) $-4x^{2}(x^{2} - 2x + 5)$

Write out the three cells of $2x(3x^{2} - x + 4)$ and the product: ____________
$-x(x^{3} - 3x^{2} + x - 9) =$ ____________
$2x(x + 3) + 5(x + 3) =$ ____________
$3x(x - 4) - 2x(x + 1) =$ ____________
Apply it. Width $6x$ m, length $x^{2} + 2x - 5$ m. Area: ____________
Apply it. Width $3x$ m, length $2x + 9$ m. Area: ____________

PAGE 19 — Practice · reasoning and errors

Practice · Degrees, Errors, and Checks

Explain. Why does the degree of a monomial times a trinomial equal the sum of the degrees? Refer to the product law.
Find the error. A student writes $3x(2x + 5) = 6x^{2} + 5$ .

Empty cell: ____________ Correct product: ____________
Find the error. A student writes $2x \cdot 4x = 8x$ .

What went wrong? _______________________________________ Correct: ____________ Law: ____________
Check 73c at $x = 2$ . Original: ______ Answer: ______
Area model: row $5x$ ; columns $x^{2}$ , $-3x$ , $2$ . Cells and product:

Cells: ______ ______ ______ Product: ____________

PAGE 20 — Exit ticket 12.4

Exit Ticket · Lesson 12.4

Name: ________________________ Date: ____________

$4x(3x^{2} - 2x + 7) =$ ____________
Area model for $-3x(x - 6)$ and the product:

_______________________________________________ Product: ____________
$x(x + 2) + 3(x + 2) =$ ____________
Check 85 at $x = -1$ . Original: ______ Answer: ______

PAGE 21 — Tile rectangle

12.5 Multiplying Two Polynomials

FIGURE: fig9-tiles-for-a-binomial-product.png (full width)

Complete the frame. If a rectangle is $x + 2$ tall and $x + 3$ wide, its area is $(x + 2)(x + 3)$ . The tiles that fill it are the ____________ of the product.

How many of each tile fill the rectangle, and what product do those tiles give?

Large: ______ Long: ______ Small: ______ Product: ____________
Name the two side lengths, and explain why the tiles that fill the rectangle are the terms of the product.

Sides: ____________ and ____________
Describe the tile rectangle for $(x + 1)(x + 4)$ — how many of each tile — and write the product.

______ large, ______ long, ______ small  Product: ____________

Sides $x + 3$ and $x + 4$ ; filled by $1$ large, $7$ long, $12$ small. Product: ____________

PAGE 22 — Two-by-two area model

Every Cell Must Be Filled

FIGURE: fig8-area-model-binomial-times-binomial.png (full width)

Three protections. The grid stops you from ____________ a pair, ____________ a sign, or forgetting to ____________ the middle cells.

Name all four cells of $(x + 3)(x - 5)$ and the product they give.

Cells: ______ ______ ______ ______ Product: ____________
Which two cells combine, and why are those two alike while the other two are not?
Multiply, showing all four cells.

Product	Four cells	Answer
a) $(x + 4)(x + 2)$
b) $(x - 3)(x + 7)$
c) $(2x + 1)(x - 5)$
d) $(3x - 2)(3x + 2)$

Multiply.

Product	Answer
a) $(x + 6)(x - 6)$
b) $(x + 5)^{2}$
c) $(2x - 3)^{2}$
d) $(x - 1)(x - 8)$

PAGE 23 — Larger grids

Binomial Times Trinomial

FIGURE: fig10-area-model-binomial-times-trinomial.png (full width)

Count the products. An $m$ -term factor times an $n$ -term factor makes exactly $m \times n$ products. A missing product is the most common error.

Name all six cells of $(2x - 3)(x^{2} + 4x - 1)$ and the product.

Cells: _______________________________________

Product: ____________
Multiply $(2x - 3)(x^{2} + 4x - 1)$ , showing all six products before you combine.

Six products: _______________________________________

Answer: ____________
$(x + 2)(x^{2} - 3x + 5) =$ ____________
$(x^{2} + 2x - 1)(x^{2} - x + 3)$ . How many products does the grid contain? ______

Product: ____________
Multiply $(x + 1)(x + 2)(x + 3)$ by multiplying two factors first, then the third.

First product: ____________ Final: ____________

PAGE 24 — Area and perimeter

Quadratic Area, Linear Perimeter

FIGURE: fig11-composite-rectangle-area-and-perimeter.png (full width)

Hold onto the contrast. Perimeter ____________ and stays ____________. Area ____________ and becomes ____________.

Give the area and the perimeter of a rectangle $x + 5$ by $x + 2$ , and explain why one is quadratic and the other is linear.

Area: ____________ Perimeter: ____________
Apply it. Patio $x + 9$ ft by $x + 4$ ft. Area: ____________ Perimeter: ____________
Apply it. Product of consecutive integers $n$ and $n + 1$ : ____________

Product of $n$ , $n + 1$ , and $n + 2$ : ____________
Find the error. A student writes $(x + 3)(x + 5) = x^{2} + 15$ .

Empty cells: ____________ Correct: ____________
Check 95c at $x = 2$ . Original: ______ Answer: ______

PAGE 25 — Exit ticket 12.5

Exit Ticket · Lesson 12.5

Name: ________________________ Date: ____________

$(x - 4)(x + 9) =$ ____________
$(3x + 1)(x^{2} - 2x + 5) =$ ____________
Area model for $(x + 2)(x + 6)$ — four cells and product:

Cells: ______ ______ ______ ______ Product: ____________
Check 107 at $x = -1$ . Original: ______ Answer: ______

PAGE 26 — Chapter 12 review · vocabulary

Chapter 12 Review

Vocabulary. polynomial · term · coefficient · degree · standard form · leading coefficient · constant term · monomial · binomial · trinomial · like terms · algebra tile · zero pair · opposite · distributive property · area model

Part A · Vocabulary and standard form

FIGURE: fig1-anatomy-of-a-polynomial.png (half width)

Write $8 - 3x^{2} + x^{4}$ in standard form: ____________

Degree: ______ Leading coefficient: ______ Constant: ______
Classify $5x^{3} - x$ by number of terms: ____________ Degree: ______
Combine: $2x^{2} - 7x + 5 + 3x - x^{2} =$ ____________
Coefficient of $x^{2}$ in $4x^{3} - x^{2} + 6$ : ______

PAGE 27 — Chapter 12 review · sums and differences

Chapter 12 Review (continued)

Part B · Sums and differences

FIGURE: fig3-adding-with-tiles.png (half width)

FIGURE: fig5-subtracting-with-tiles.png (half width)

$(3x^{2} - 8x + 1) + (x^{2} + 8x - 10) =$ ____________
$(6x^{2} + x - 4) - (2x^{2} - 5x + 3) =$ ____________
Collection: $2$ large, $3$ long, $1$ small. Take away $1$ large and $4$ long.

Write the subtraction: ____________

Zero pairs: _______________________________________

Result: ____________
Find the error. A student writes $(5x^{2} + 2x) - (3x^{2} - x) = 2x^{2} + x$ .

What went wrong? _______________________________________ Correct: ____________

PAGE 28 — Chapter 12 review · products

Chapter 12 Review (continued)

Part C · Products

FIGURE: fig7-area-model-monomial-times-trinomial.png (half width)

FIGURE: fig8-area-model-binomial-times-binomial.png (half width)

$-5x(2x^{2} - x + 3) =$ ____________
$(x + 7)(x - 2) =$ ____________
$(2x - 1)(x^{2} + 3x - 4)$ , showing all six products:

Answer: ____________
Area model for $(3x + 2)(x + 5)$ — four cells and product:

Cells: ______ ______ ______ ______ Product: ____________

PAGE 29 — Chapter 12 review · models both ways

Chapter 12 Review (continued)

Part D · Models in both directions

FIGURE: fig4-zero-pairs-cancel.png (half width)

FIGURE: fig9-tiles-for-a-binomial-product.png (half width)

FIGURE: fig10-area-model-binomial-times-trinomial.png (half width)

Collection: $1$ large, $2$ long, $6$ small. Polynomial: ____________

Tiles for its opposite: _______________________________________
Tile rectangle sides $x + 5$ and $x + 2$ . How many of each tile, and the product?

______ large, ______ long, ______ small Product: ____________
Tile picture for $(2x^{2} + x + 4) - (x^{2} + 3x + 1)$ :

Starting: _______________________________________

Opposite added: _______________________________________

Zero pairs: _______________________________________

Left: _______________________________________ Difference: ____________
Rows $x$ and $-4$ ; columns $x^{2}$ , $2x$ , $-3$ . Six cells and product:

Cells: _______________________________________

Product: ____________

PAGE 30 — Chapter 12 review · mixed application

Chapter 12 Review (continued)

Part E · Mixed application

FIGURE: fig11-composite-rectangle-area-and-perimeter.png (full width)

FIGURE: fig2-like-terms-sorting.png (half width)

FIGURE: fig6-distribute-the-minus-sign.png (half width)

Apply it. Garden $2x + 3$ m by $x + 6$ m.

Area: ____________ Perimeter: ____________

Which is quadratic, and why? _______________________________________
Apply it. $R(x) = x(150 - 4x)$ , $C(x) = 30x + 200$ .

Expand $R(x)$ : ____________ Profit $R - C$ : ____________
Apply it. Product of $n$ and $n + 2$ : ____________

Sum of their squares: ____________
Apply it. Rectangle $x + 8$ by $x + 5$ ; square of side $x$ removed.

Remaining area: ____________

Check at $x = 2$ : original area ______ − square ______ = ______; polynomial ______

PAGE 31 — Blank tile mats and area grids

Build the Picture Yourself

FIGURE: fig12-blank-tile-mats-and-area-grids.png (full page)

Use these blanks for any tile or area-model exploration a teacher assigns alongside this chapter.

Mat A and Mat B. Draw the tiles for a polynomial the teacher names, then write the polynomial on the line beneath. Or work the other direction: write a polynomial first, then draw the tiles that match it. Use shaded tiles for positive terms and white tiles for negatives. Mark every zero pair you remove with an X.

Grid A — binomial times binomial. Label the two rows and two columns, fill all four cells, combine like terms, and write the product beneath the grid.

Grid B — binomial times trinomial. Label the two rows and three columns, fill all six cells, combine, and write the product. Count: six products before combining — if you have fewer, a cell is empty.

Two reminders for every blank you fill:

Every cell must be filled. A missing middle cell is the classic error of $(x + 3)(x + 5) = x^{2} + 15$ .
The sign sits on the label. A column labeled $-5$ already carries its minus into every cell in that column.

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Item order: a few items are placed by figure rather than by number, so that every item on a page can be answered from the figure on that page. Page 3 runs 5, 6, 11, 12; page 7 runs 23–25, 27, 28; page 12 runs 47, 48, 54; page 21 runs 89, 90, 101, 102; page 22 runs 91, 92, 95, 96. The numbers still match the textbook exactly, and the answer key is in numerical order.
Figure widths: fig1, fig2, fig3, fig4, fig5, fig6, fig7, fig8, fig9, fig10, and fig11 are full width on first appearance. fig5-subtracting-with-tiles.png, fig6-distribute-the-minus-sign.png, and fig10-area-model-binomial-times-trinomial.png are wide multi-panel images — set them full width and do not place text beside them. fig12-blank-tile-mats-and-area-grids.png takes a full page on its own. fig1, fig2, fig3, fig4, fig5, fig6, fig7, fig8, fig9, fig10, and fig11 repeat in the review; repeat the image rather than asking students to flip back.
Blank work space: items 3, 6, 12, 16, 25, 37, 38, 45, 46, 49, 50, 58, 59, 62, 66, 69, 80, 81, 82, 90, 92, 94, 105, 117, 118, 123, 125, and 127 are written explanations — give each at least four ruled lines, not a single blank. Items 14, 18, 33, 34, 35, 36, 56, 57, 60, 78, 79, 103, 104, 127, 128, 129, and 130 are applications with two or more parts; leave two lines per part.
Negative signs: use the typographic minus (−), matching the figures, not a hyphen. This matters for every subtraction and every negative coefficient.
No factoring, no division, no solving. This chapter only simplifies expressions. If a proof copy shows a "factor" prompt, a long-division setup, or an equation to solve, it is a paste error from Chapters 13–15 and must come out.
One variable only; factors ≤ 5 terms. Every polynomial is in $x$ (or $n$ , $t$ in context). The largest grids are $2 \times 3$ and $3 \times 3$ .