Algebra 1 Workbook — Chapter 14: Dividing Polynomials and Equivalent Quadratic Forms

SOL A.EO.2 (d, e) · Companion to Textbook Chapter 14

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 110.

PAGE 1 — Chapter opener

Chapter 14 · Dividing Polynomials and Equivalent Quadratic Forms

Standard A.EO.2 (d, e)

In this chapter you will:

Divide a polynomial by a monomial, term by term
Divide by a binomial with an area model, then multiply back to check
Cancel a completely factored divisor and state the domain restriction
Rewrite $x^{2} + bx$ by completing the square (expression rewrite only)
Write one quadratic in standard, factored, and vertex form, and prove they are equal by expanding

Words to know: dividend · divisor · quotient · monomial divisor · binomial divisor · completely factored divisor · domain restriction · cancellation · area model · multiply-back check · standard form · factored form · vertex form · completing the square · equivalent forms

Convention: every cancellation carries a restriction. Canceling $(x - a)$ requires $x \neq a$ . The simplified expression and the original agree everywhere the original exists.

Convention: vertex form is algebraic. $a(x - h)^{2} + k$ reveals the least value $k$ (when $a > 0$ ) at $x = h$ — without drawing a parabola.

Convention: expanding proves equality. A table of matching outputs is evidence, not proof.

Calculator. Algebra 1 has no no-calculator standards. Use one to confirm: evaluate the original and your answer at an allowed input.

PAGE 2 — Monomial division, term by term

14.1 Dividing by a Monomial

FIGURE: fig1-monomial-divisor-term-by-term.png (full width)

Fill in the blanks.

A monomial divisor splits the dividend into ____________ separate fractions.

In each fraction, ____________ the coefficients and ____________ the exponents.

The rewriting is valid for every $x \neq$ ______, because the original expression does not exist at that input.

Write the three separate fractions for $\dfrac{12x^{4} + 18x^{3} - 6x^{2}}{3x^{2}}$ .
First row: coefficients ______ $\div$ ______ $=$ ______; exponents ______ $-$ ______ $=$ ______; quotient term ______
Why does the third row produce the constant $-2$ ?
Restriction: $x \neq$ ______ Why required? _______________________________________

PAGE 3 — Practice · dividing by a monomial

Practice · One Fraction per Term

$\dfrac{15x^{5} - 10x^{3} + 5x}{5x} =$ ______ Restriction: ______

Check: $5x \cdot ($ ______ $) =$ ______
Multiply-back check for item 5 shown above: ______
Divide. State the restriction when needed.

Division	Quotient	Restriction
a) $\dfrac{18x^{4} + 27x^{3} - 9x^{2}}{9x^{2}}$
b) $\dfrac{10x^{3} - 15x^{2} + 5x}{5x}$
c) $\dfrac{24x^{5} - 16x^{3}}{8x}$
d) $\dfrac{14x^{4} + 21x^{2}}{7x^{2}}$

Divide.

Division	Quotient	Restriction
a) $\dfrac{12x^{3} - 18x^{2} + 6x}{6x}$
b) $\dfrac{-20x^{6} + 10x^{4} - 30x^{2}}{5x^{2}}$
c) $\dfrac{9x^{4} - 3x^{2} + 6}{3}$
d) $\dfrac{16x^{5} - 4x^{3}}{4x^{3}}$

PAGE 4 — Practice · restrictions and checks

Practice · Restrictions and Multiply-Back

$\dfrac{x^{4} - 5x^{3} + 2x}{x} =$ ______ Restriction: ______
$\dfrac{25x^{4} - 10x^{3} + 15x^{2}}{5x^{2}} =$ ______ Restriction: ______
Explain. Why are $\dfrac{12x^{4} + 18x^{3} - 6x^{2}}{3x^{2}}$ and $4x^{2} + 6x - 2$ not identical as expressions?
Error analysis. Student writes $\dfrac{8x^{3} - 12x^{2} + 4x}{4x} = 2x^{2} - 3x + 4x$ . Error: ____________ Correct: ______
Error analysis. Student writes $\dfrac{15x^{5} - 10x^{3}}{5x} = 3x^{4} - 2x^{3}$ . Error: ____________ Correct: ______
Check $\dfrac{18x^{4} + 27x^{3} - 9x^{2}}{9x^{2}} = 2x^{2} + 3x - 1$ by multiplying back:

$9x^{2}(2x^{2} + 3x - 1) =$ ______

PAGE 5 — Practice · applications and exit 14.1

Practice · Context and Exit Ticket 14.1

Application. Volume $= 24x^{3} + 36x^{2}$ cm³, height $= 12x$ cm. Base area: ______ Restriction: ______
Evaluate both $\dfrac{12x^{4} + 18x^{3} - 6x^{2}}{3x^{2}}$ and $4x^{2} + 6x - 2$ at $x = 2$ .

Original: ______ Quotient: ______ Why not $x = 0$ ? ______
$\dfrac{-8x^{4} + 12x^{3} - 4x^{2}}{-4x^{2}} =$ ______
Invent: dividend ______, divisor ______, quotient $5x^{2} - 3$ , restriction $x \neq 0$ .
$\dfrac{27x^{6} - 18x^{4} + 9x^{2}}{9x^{2}} =$ ______
Explain. Why does a constant divisor need no $x \neq 0$ restriction, while $3x^{2}$ does?

Exit ticket 14.1

$\dfrac{12x^{4} + 18x^{3} - 6x^{2}}{3x^{2}} =$ ______ Restriction: ______
$\dfrac{20x^{5} - 15x^{3} + 10x}{5x} =$ ______
Check item 22: $5x \cdot ($ ______ $) =$ ______
Why does the quotient law subtract exponents? _______________________________________

PAGE 6 — Area model for binomial division

14.2 Dividing by a Binomial

FIGURE: fig2-area-model-for-division.png (full width)

Fill in the blanks.

The top edge is the ____________. The four cells add to the ____________. The left edge is the ____________.

Divisor: ______ Four cells: ______, ______, ______, ______
How is the left edge forced? Top row: ______ $\div$ ______ $=$ ______. Bottom: ______ $\div$ ______ $=$ ______.

PAGE 7 — Multiply-back check

Checking by Multiplying Back

FIGURE: fig4-checking-a-division-by-multiplying-back.png (full width)

$(3x - 1)(2x + 3) =$ ______ $=$ ______ ✓
$\dfrac{x^{2} + 5x + 6}{x + 2} =$ ______ Restriction: ______
$\dfrac{x^{2} - 9}{x + 3} =$ ______ Restriction: ______
Why is the restriction for $\dfrac{6x^{2} + 7x - 3}{3x - 1}$ equal to $x \neq \tfrac13$ , not $x \neq 0$ ?

PAGE 8 — Practice · binomial quotients

Practice · Binomial Divisors

Divide. State the restriction.

Division	Quotient	Restriction
a) $\dfrac{x^{2} + 9x + 20}{x + 4}$
b) $\dfrac{x^{2} - 5x + 6}{x - 2}$
c) $\dfrac{x^{2} + 5x + 6}{x + 2}$
d) $\dfrac{x^{2} - 16}{x - 4}$

Divide.

Division	Quotient	Restriction
a) $\dfrac{2x^{2} + 7x + 3}{x + 3}$
b) $\dfrac{2x^{2} + 11x + 12}{x + 4}$
c) $\dfrac{6x^{2} + 7x - 3}{3x - 1}$
d) $\dfrac{3x^{2} - 5x - 2}{3x + 1}$

$\dfrac{2x^{2} - x - 3}{x + 1} =$ ______ Multiply-back: ______
$\dfrac{4x^{2} - 4x - 3}{2x + 1} =$ ______ Restriction: ______

PAGE 9 — Blank area models

Practice · Blank Area Models

FIGURE: fig8-blank-area-models-for-division.png (full width)

Fill all three models. For each: top-edge labels, left-edge quotient, four cells.

Left: quotient ______ Cells: _______________________

Middle: quotient ______ Cells: _______________________

Right: quotient ______ Cells: _______________________
Middle model check: sum of cells $=$ ______ (should equal $2x^{2} + 11x + 12$ )
Application. Area $x^{2} + 9x + 20$ m², width $x + 4$ m. Length: ______ Restriction: ______
Application. Revenue $(3x - 1)(2x + 3)$ dollars. Expanded: ______ Divide by $3x - 1$ : ______ Restriction: ______

PAGE 10 — Practice · errors and exit 14.2

Practice · Errors and Exit Ticket 14.2

Error analysis. Student: $\dfrac{x^{2} + 9x + 20}{x + 4} = x + 4$ . Error: ______ Correct: ______ Check: ______
Error analysis. Student: $\dfrac{2x^{2} + 7x + 3}{x + 3} = 2x + 3$ . Multiply-back shows: ______ Correct: ______
$\dfrac{x^{2} - 8x + 15}{x - 3} =$ ______
$\dfrac{5x^{2} + 14x - 3}{x + 3} =$ ______
Check: $(3x + 1)(x - 2) =$ ______
Explain. Why must area-model cells add to the dividend?
At $x = 1$ : $\dfrac{2x^{2} + 7x + 3}{x + 3} =$ ______ and $2x + 1 =$ ______
$\dfrac{x^{2} + 7x + 12}{x + 3} =$ ______ Restriction: ______

Exit ticket 14.2

$\dfrac{2x^{2} + 7x + 3}{x + 3} =$ ______ Restriction: ______
$\dfrac{6x^{2} + 7x - 3}{3x - 1} =$ ______ Multiply-back: ______
$\dfrac{x^{2} + 9x + 20}{x + 4} =$ ______
Why does multiplying quotient by divisor check a division? _______________________________________

PAGE 11 — Cancellation and domain restrictions

14.3 Completely Factored Divisors and Domain Restrictions

FIGURE: fig3-cancellation-with-a-domain-restriction.png (full width)

Fill in the blanks.

$\dfrac{(x - 3)(x + 1)}{x - 3} =$ ______ provided $x \neq$ ______.

At $x = 3$ the left side is ____________, while the right side is ______.

Write the four steps shown in the figure.
$\dfrac{x - 3}{x - 3} = 1$ only when ______, because ______.
At $x = 3$ : left ______, right ______. At $x = 4$ : left ______, right ______.
$\dfrac{(x + 2)(x - 5)}{x + 2} =$ ______ Restriction: ______
$\dfrac{(x + 3)(x - 2)(x + 1)}{(x + 3)(x - 2)} =$ ______ Restrictions: ______
Factor then cancel: $\dfrac{x^{2} + 9x + 20}{x + 4} =$ ______ Restriction: ______

PAGE 12 — Practice · cancellation

Practice · Cancel and Restrict

Simplify. State every restriction.

Expression	Simplified	Restriction(s)
a) $\dfrac{(x - 3)(x + 1)}{x - 3}$
b) $\dfrac{(x + 2)(x - 5)}{x + 2}$
c) $\dfrac{(2x - 1)(x + 4)}{2x - 1}$
d) $\dfrac{(x - 4)(x + 4)}{x - 4}$

Simplify.

Expression	Simplified	Restriction(s)
a) $\dfrac{(x + 1)(x + 1)}{x + 1}$
b) $\dfrac{(x - 2)(x + 3)(x - 5)}{(x - 2)(x + 3)}$
c) $\dfrac{(3x + 1)(x - 2)}{3x + 1}$
d) $\dfrac{(x + 6)(2x - 3)}{x + 6}$

$\dfrac{x^{2} - 5x + 6}{x - 2} =$ ______ (factor first)
$\dfrac{x^{2} - 16}{x + 4} =$ ______ (factor first)

PAGE 13 — Practice · why restrictions matter

Practice · Restrictions Matter

Explain. Student writes $x + 1$ with no restriction for $\dfrac{(x - 3)(x + 1)}{x - 3}$ . What is missing, and why does it matter?
Error analysis. Student cancels the $x$ in $\dfrac{x + 5}{x}$ and writes $5$ . Error: ______
Error analysis. Student writes $\dfrac{(x - 1)(x - 1)}{x - 1} = 1$ . Error: ______ Correct: ______
At $x = 0$ : both sides ______. At $x = 5$ : both sides ______. Why not check at $x = 3$ ? ______
Application. Profit density $\dfrac{(x - 2)(x + 5)}{x - 2}$ . Simplified: ______ Undefined at $x =$ ______
$\dfrac{(x + 4)(x - 1)(x + 2)}{(x + 4)(x + 2)} =$ ______ Restrictions: ______
Factor then cancel: $\dfrac{6x^{2} + 7x - 3}{3x - 1} = \dfrac{(3x - 1)(2x + 3)}{3x - 1} =$ ______
Explain. Why must the divisor be completely factored before you cancel?
$\dfrac{(5x - 2)(x + 7)}{5x - 2} =$ ______ Restriction: ______
Invent an expression that simplifies to $x - 5$ with restriction $x \neq 2$ : ______

Exit ticket 14.3

$\dfrac{(x - 3)(x + 1)}{x - 3} =$ ______ Restriction: ______
$\dfrac{(x + 3)(x - 2)(x + 1)}{(x + 3)(x - 2)} =$ ______ Restrictions: ______
$\dfrac{x^{2} + 5x + 6}{x + 2} =$ ______ Restriction: ______
Why are the simplified form and the original not the same at the restricted input?

PAGE 14 — Completing the square

14.4 Equivalent Quadratic Forms

FIGURE: fig5-completing-the-square-area-picture.png (full width)

Missing corner for $x^{2} + 8x$ : $\left(\dfrac{8}{2}\right)^{2} =$ ______. Why write $-16$ after $(x + 4)^{2}$ ?
$x^{2} + 6x = ($ ______ $)^{2} -$ ______
$x^{2} + 10x = ($ ______ $)^{2} -$ ______
$x^{2} - 4x = ($ ______ $)^{2} -$ ______
Expand: $(x - 2)^{2} - 1 =$ ______ (should equal $x^{2} - 4x + 3$ )

PAGE 15 — Three forms of one quadratic

One Expression, Three Forms

FIGURE: fig6-three-forms-of-one-quadratic.png (full width)

Three expressions: ______ $=$ ______ $=$ ______

Expand factored → standard: ______
Expand vertex → standard: ______

FIGURE: fig7-two-forms-agree-in-a-table.png (full width)

At $x = 3$ , all three forms give ______. What does that tell you about vertex form?

PAGE 16 — Blank completing-the-square frames

Practice · Completing the Square

FIGURE: fig9-blank-completing-the-square-frames.png (full width)

Rewrite by completing the square.

Expression	Vertex form
a) $x^{2} + 6x$
b) $x^{2} + 10x$
c) $x^{2} - 4x$
d) $x^{2} - 6x$

Fill the blank frames: corner added and vertex form for each of the three expressions.

Left ( $x^{2} + 6x$ ): corner ______ Form ______

Middle ( $x^{2} + 10x$ ): corner ______ Form ______

Right ( $x^{2} - 4x$ ): corner ______ Form ______
Rewrite in vertex form.

Standard	Vertex
a) $x^{2} - 6x + 5$
b) $x^{2} - 4x + 3$
c) $x^{2} + 2x - 8$
d) $x^{2} - 8x + 12$

PAGE 17 — Proving equality

Practice · Expanding Proves Equality

Expand to standard form.

Expression	Standard form
a) $(x - 1)(x - 5)$
b) $(x - 3)^{2} - 4$
c) $(x + 4)(x - 2)$
d) $(x + 1)^{2} - 9$

For $x^{2} - 4x + 3$ : factored ______ vertex ______ Expansions prove: ______
Use the blank three-form panels for $x^{2} - 6x + 5$ .

FIGURE: fig10-blank-three-form-panels.png (full width)

Standard: ______  Reveals: ______

Factored: ______  Reveals: ______

Vertex: ______  Reveals: ______

Expansion proof: ______

PAGE 18 — Context, tables, errors

Practice · Context and Reasoning

Application. Height expression $-16t^{2} + 32t + 48$ . Factor $-16$ from the first two terms, complete the square on $t^{2} - 2t$ , and write vertex form. Greatest value: ______ at $t =$ ______
Table at $x = -4, -1, 0, 2$ for $x^{2} + 2x - 8$ , $(x + 4)(x - 2)$ , $(x + 1)^{2} - 9$ . Rows agree? ______ Expand to prove: ______
Explain. Why is a table evidence but not proof?
Error analysis. Student writes $x^{2} + 8x = (x + 4)^{2}$ . Error: ______ Correct: ______
Error analysis. Student claims $x^{2} - 6x + 5 = (x - 3)^{2} + 4$ . Expand second: ______ Correct vertex: ______
Expand $2(x - 2)^{2} - 2 =$ ______. Least value ______ at $x =$ ______
$x^{2} - 8x + 12$ in three forms: ______ $=$ ______ $=$ ______

What each reveals: _______________________________________
Explain. From $a(x - h)^{2} + k$ with $a > 0$ , why is the least value $k$ at $x = h$ , with no parabola drawn?

Exit ticket 14.4

$x^{2} + 8x =$ ______
$x^{2} - 6x + 5 =$ ______ $=$ ______ $=$ ______
Expand both $(x - 3)^{2} - 4$ and $(x - 1)(x - 5)$ : ______
What does $(x - 3)^{2} - 4$ reveal that standard form does not? ______

PAGE 19 — Chapter 14 review · division

Chapter 14 Review

$\dfrac{12x^{4} + 18x^{3} - 6x^{2}}{3x^{2}} =$ ______ Restriction: ______
$\dfrac{-20x^{6} + 10x^{4} - 30x^{2}}{5x^{2}} =$ ______
Check item 101: $3x^{2} \cdot ($ ______ $) =$ ______
$\dfrac{2x^{2} + 7x + 3}{x + 3} =$ ______ Restriction: ______
$\dfrac{6x^{2} + 7x - 3}{3x - 1} =$ ______ Multiply-back: ______
$\dfrac{x^{2} + 9x + 20}{x + 4} =$ ______
$\dfrac{(x - 3)(x + 1)}{x - 3} =$ ______ Restriction: ______
$\dfrac{(x + 3)(x - 2)(x + 1)}{(x + 3)(x - 2)} =$ ______ Restrictions: ______

PAGE 20 — Chapter 14 review · forms

Chapter 14 Review (continued)

$x^{2} - 6x + 5$ factored: ______ vertex: ______

Expand factored: ______  Expand vertex: ______

$x^{2} + 10x = ($ ______ $)^{2} -$ ______ Least value ______ at $x =$ ______

PAGE 21 — Production notes

Canva production notes

Paste each PAGE section as one Canva page; keep the FIGURE: lines as image placeholders and drop in the matching PNG from ../figures/.
Full-width figures: fig1, fig2, fig3, fig4, fig5, fig6, fig7, fig8, fig9, fig10.
Leave generous write-on lines for restrictions — students must record them, not only the simplified expression.
Item numbers match the textbook exactly (1–110).