Algebra 1 Workbook — Chapter 13: Factoring Polynomials

SOL A.EO.2 (c) · Companion to Textbook Chapter 13

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 132.

PAGE 1 — Chapter opener

Chapter 13 · Factoring Polynomials

Standard: A.EO.2 (c)

In this chapter you will:

Treat factoring as Chapter 12's multiplication run backwards
Factor out the GCF, including a leading minus sign
Factor $x^{2} + bx + c$ by a factor-pair search, and say prime when the search is exhausted
Factor a difference of squares and a perfect-square trinomial
Factor $ax^{2} + bx + c$ by splitting the middle term and grouping
Factor completely and choose a method from the shape of the polynomial

Words to know: factor · factor completely · greatest common factor (GCF) · prime over the integers · factor pair · difference of squares · perfect-square trinomial · split the middle term · grouping · area model

Convention: factoring rewrites an expression as a product. It does not solve an equation. No item in this chapter asks for a value of $x$ .

Convention: GCF first, every time. After the GCF, the remaining leading coefficient has at most four factors.

Convention: check by multiplying the factors back out. Evaluating at $x = 2$ and $x = -1$ is a second check.

Calculator. Use one to confirm a factorization you already produced — not to invent the factors.

PAGE 2 — Multiplying and factoring are reverses

13.1 Factoring as Multiplication Run Backwards

FIGURE: fig1-multiplying-and-factoring-are-reverses.png (full width)

Fill in the blanks.

On the multiplying side, the sides are ______ and ______ across the top, ______ and ______ down the side.

The four cells are ______, ______, ______, and ______. They sum to ______.

On the factoring side, the ______ are given and the ______ are blank.

Four cells of $(x + 3)(x + 2)$ : _______________ Polynomial: ______
What the factoring panel asks: _______________________________________________

PAGE 3 — Area model run backwards

Practice · Reading Cells, Finding Sides

FIGURE: fig2-area-model-run-backwards.png (full width)

Four cells: _______________ Polynomial: ______
How $2x$ and $4x$ make the middle term $6x$ : _______________________________________________
Side lengths: _______________ Factored form: ______
Explain. Why is factoring not a new operation?
Cells $x^{2}$ , $4x$ , $2x$ , $8$ → polynomial ______ sides ______ factorization ______
Cells $x^{2}$ , $5x$ , $3x$ , $15$ → factorization ______

PAGE 4 — Practice · reverse and check

Practice · Writing Factorizations and Checking

Multiply $(x + 1)(x + 6)$ = ______ Then write ______ = $(x + 1)(x + 6)$ .
Multiply $(x - 2)(x + 7)$ = ______ Factorization: ______
Check $x^{2} + 9x + 18 = (x + 3)(x + 6)$ by multiplying back: ______ Correct? ______
Check $x^{2} + 8x + 12 = (x + 2)(x + 6)$ at $x = 2$ : ______ at $x = -1$ : ______
Error analysis. Cells $x^{2}$ , $2x$ , $4x$ , $8$ with claim $(x + 2)(x + 2)$ . Error: _______________ Correct: ______
Application. Area $x^{2} + 7x + 10$ . Length and width: _______________
Reasoning. Why is multiplying back a complete check, while one evaluation is not?

PAGE 5 — Exit ticket 13.1

Exit Ticket · 13.1

$x^{2} + 5x + 6 =$ ______
Cells $x^{2}$ , $2x$ , $4x$ , $8$ → sides ______
Multiply your factors from 16 back out: ______
One sentence: what "factoring is multiplication run backwards" means for an area model.

PAGE 6 — GCF as a shared side

13.2 The Greatest Common Factor

FIGURE: fig3-gcf-as-a-shared-side.png (full width)

Why is there only one row of cells? _______________________________________________
Shared side: ______ Divisions: $6x^{2} \div$ ______ $=$ ______ and $15x \div$ ______ $=$ ______
Multiply $3x(2x + 5)$ back: ______
GCF of $10x^{2} - 15x$ : ______ Factored: ______
$-6x^{2} + 9x =$ ______
Explain. Why does GCF come before every other method?

PAGE 7 — Practice · GCF

Practice · Pulling Out a Monomial

Factor.

Expression	Factored
a) $4x + 12$
b) $9x - 15$
c) $10x + 25$
d) $14x - 21$

Factor.

Expression	Factored
a) $6x^{2} + 9x$
b) $10x^{2} - 15x$
c) $8x^{2} + 12x$
d) $5x^{2} - 20x$

Factor (take the minus with the GCF when needed).

Expression	Factored
a) $-3x + 12$
b) $-4x^{2} + 10x$
c) $-6x^{2} - 9x$
d) $-x^{2} + 5x$

$12x^{2} + 18x =$ ______ Check: ______
$7x^{2} + 7x =$ ______ Remaining factor is a ______

PAGE 8 — Practice · GCF in context

Practice · Incomplete GCF and Context

Error analysis. $6x^{2} + 15x = 3(2x^{2} + 5x)$ . Missed: _______________ Correct: ______
Application. Area $8x^{2} + 20x$ . Factored: ______ Possible length ______ width ______
Reasoning. Why is $2(x^{2} - 4)$ correct but not complete?
Expand $5x(2x - 3) =$ ______ Original polynomial: ______
$6x^{2} + 15x =$ ______
$-4x^{2} + 12x =$ ______
$9x + 6 =$ ______ Check: ______
The shared side in the GCF model represents: _______________________________________________

PAGE 9 — Factor-pair search

13.3 Trinomials with Leading Coefficient 1

FIGURE: fig4-factor-pair-search-table.png (full width)

Matching row for $x^{2} + 11x + 24$ : ______ Why: _______________
Sums on the right table: _______________ Why prime: _______________

FIGURE: fig11-blank-factor-pair-tables.png (full width)

Use Table A for item 43. Product $=$ ______ Sum $=$ ______

Factor pairs of $18$ and sums:

Factor pair	Sum

Factored form: $x^{2} + 9x + 18 =$ ______

$x^{2} - 7x + 12 =$ ______

PAGE 10 — Tiles versus prime

Practice · Rectangle or Prime

FIGURE: fig5-tile-rectangle-versus-prime.png (full width)

Left rectangle: ______ $x^{2}$ tiles, ______ $x$ tiles, ______ unit tiles. Product: ______
Why $x^{2} + 5x + 8$ cannot form a rectangle: _______________________________________________
Factor.

Trinomial	Factors
a) $x^{2} + 8x + 15$
b) $x^{2} + 9x + 20$
c) $x^{2} + 12x + 32$
d) $x^{2} + 10x + 21$

Factor.

Trinomial	Factors
a) $x^{2} - 5x + 6$
b) $x^{2} - 9x + 20$
c) $x^{2} - 8x + 12$
d) $x^{2} - 11x + 24$

PAGE 11 — Practice · signs and primes

Practice · Mixed Signs and Primes

Use Tables B and C from fig11-blank-factor-pair-tables.png for 47a and 48a.

Factor.

Trinomial	Factors
a) $x^{2} + x - 12$
b) $x^{2} - x - 12$
c) $x^{2} + 3x - 10$
d) $x^{2} - 2x - 15$

Factorable or prime?

Trinomial	Answer
a) $x^{2} + 7x + 15$
b) $x^{2} + 5x + 8$
c) $x^{2} + 6x + 8$
d) $x^{2} + 4x + 8$

$x^{2} + 11x + 24 =$ ______ Check: ______
Table for $x^{2} - 3x - 10$ : product ______ sum ______ Factors: ______
Error analysis. $(x + 1)(x + 5)$ for $x^{2} + 5x + 6$ . Error: _______________ Correct: ______
Application. Area $x^{2} + 9x + 18$ . Length ______ width ______

PAGE 12 — Exit ticket 13.3

Exit Ticket · 13.3

Reasoning. Why does listing every factor pair make "prime" a conclusion?
Expand $(x - 4)(x - 3) =$ ______ Matches which lesson trinomial? ______
$x^{2} + 11x + 24 =$ ______
$x^{2} - 5x + 6 =$ ______
Is $x^{2} + 5x + 8$ factorable? Justify: _______________________________________________
$x^{2} + x - 12 =$ ______ Check: ______

PAGE 13 — Difference of squares

13.4 Difference of Squares

FIGURE: fig6-difference-of-squares-area.png (full width)

Removed: ______ Pieces remaining: _______________
Rearranged sides: ______ and ______ Identity: ______
$x^{2} - 16$ : $a =$ ______ $b =$ ______ Factored: ______
$4x^{2} - 9 =$ ______
$2x^{2} - 8$ completely: ______
Explain. Why $x^{2} + 9$ is not a difference of squares: _______________________________________________

PAGE 14 — Practice · difference of squares

Practice · $a^{2} - b^{2}$

Factor.

Expression	Factored
a) $x^{2} - 9$
b) $x^{2} - 16$
c) $x^{2} - 25$
d) $x^{2} - 36$

Factor.

Expression	Factored
a) $4x^{2} - 9$
b) $9x^{2} - 25$
c) $16x^{2} - 1$
d) $25x^{2} - 49$

Factor completely.

Expression	Factored
a) $2x^{2} - 8$
b) $3x^{2} - 27$
c) $5x^{2} - 45$
d) $2x^{2} - 32$

Difference of squares, prime, or not a difference of squares?

Expression	Answer
a) $x^{2} - 4$
b) $x^{2} + 4$
c) $x^{2} - 2$
d) $9x^{2} - 16$

PAGE 15 — Practice · difference in context

Practice · Checks and Context

$9x^{2} - 25 =$ ______ Check: ______
Error analysis. $x^{2} - 9 = (x - 3)^{2}$ . Error: _______________ Correct: ______
Application. Remaining area $x^{2} - 9$ . As a product: ______
Reasoning. Why $(a + b)(a - b) = a^{2} - b^{2}$ (middle terms cancel):
Expand $(2x + 5)(2x - 5) =$ ______ Reverses which item? ______
$x^{2} - 9 =$ ______
$4x^{2} - 9 =$ ______
$2x^{2} - 8$ completely: ______
Why $x^{2} + 9$ is not a difference of squares: _______________________________________________

PAGE 16 — Perfect-square trinomials

13.5 Perfect Squares and $a > 1$

FIGURE: fig7-perfect-square-trinomial.png (full width)

Left panel: equal sides ______ Written ______
Right panel cells: _______________ How middle cells make $-20x$ : _______________

FIGURE: fig10-blank-area-models.png (full width)

Use Model A for $x^{2} + 6x + 9$ . Fill sides and cells.

$x^{2} - 8x + 16 =$ ______
Factor as a perfect square.

Expression	Factored
a) $x^{2} + 2x + 1$
b) $x^{2} + 10x + 25$
c) $x^{2} - 8x + 16$
d) $x^{2} - 12x + 36$

Factor as a perfect square.

Expression	Factored
a) $4x^{2} + 12x + 9$
b) $4x^{2} - 20x + 25$
c) $9x^{2} + 6x + 1$
d) $9x^{2} - 30x + 25$

PAGE 17 — Splitting the middle term

Practice · Split and Group

FIGURE: fig8-splitting-the-middle-term.png (full width)

Why $7x = 6x + x$ : _______________________________________________
Shared binomial: ______ Factored: ______

Use Model B from fig10-blank-area-models.png for $2x^{2} + 7x + 3$ .

$2x^{2} + 11x + 12 =$ ______ (show the split)
Factor by splitting.

Expression	Split	Factored
a) $2x^{2} + 7x + 3$
b) $2x^{2} + 11x + 12$
c) $3x^{2} + 10x + 8$
d) $3x^{2} + 11x + 6$

Factor by splitting.

Expression	Factored
a) $6x^{2} + 5x - 6$
b) $6x^{2} - x - 2$
c) $4x^{2} + 5x - 6$
d) $2x^{2} - 5x - 3$

PAGE 18 — Practice · squares and $a > 1$

Practice · Two Methods and Context

$4x^{2} + 12x + 9$ as a square: ______ by splitting: ______ Match? ______
Error analysis. $(x + 5)^{2} = x^{2} + 25$ . Missing: ______ Correct expansion: ______ Factor $x^{2} + 10x + 25$ : ______
Application. Area $x^{2} + 6x + 9$ . Side length: ______
Reasoning. Why search pairs of $ac$ , not $c$ ?
Group $2x^{2} + 6x + x + 3$ : ______
$6x^{2} + 5x - 6 =$ ______ Check: ______
$x^{2} + 6x + 9 =$ ______
$4x^{2} - 20x + 25 =$ ______
$2x^{2} + 7x + 3 =$ ______
$6x^{2} - x - 2 =$ ______

PAGE 19 — Choosing a method

13.6 Factoring Completely and Choosing a Method

FIGURE: fig9-choosing-a-method-flow.png (full width)

Step 1: _______________ Why first: _______________
Step 2 branches: _______________ / _______________ / _______________
When is prime the correct final answer? _______________________________________________
$2x^{2} - 8$ : Step 1 ______ → Step 2 ______ → complete ______
$3x^{2} + 6x + 3 =$ ______
Explain. Why end with a multiply-back check?

_______________________________________________

PAGE 20 — Practice · factor completely

Practice · Complete Factorizations

Factor completely.

Expression	Factored
a) $2x^{2} - 8$
b) $3x^{2} - 27$
c) $5x^{2} - 20$
d) $4x^{2} - 36$

Factor completely.

Expression	Factored
a) $3x^{2} + 6x + 3$
b) $2x^{2} + 8x + 8$
c) $5x^{2} + 10x + 5$
d) $4x^{2} + 4x + 1$

Completely, or prime.

Expression	Answer
a) $x^{2} + 7x + 15$
b) $x^{2} + 5x + 6$
c) $x^{2} + 5x + 8$
d) $x^{2} - 9$

First method after GCF, then factor completely.

Expression	First method	Complete
a) $x^{2} - 25$
b) $x^{2} + 8x + 16$
c) $2x^{2} + 7x + 3$
d) $6x^{2} + 15x$

PAGE 21 — Practice · flow and context

Practice · The Flow in Context

Use Models C and D from fig10-blank-area-models.png for 109 and one review item.

Error analysis. $2x^{2} - 8 = 2(x^{2} - 4)$ and stop. Catch: _______________ Complete: ______
Application. Area $2x^{2} + 7x + 3$ . Factored: ______ Sides: _______________
Reasoning. From $2(x + 3)(x - 3)$ , recover the original and explain completeness:
$6x^{2} + 5x - 6 =$ ______ At $x = 2$ : original ______ factors ______
$-2x^{2} + 8$ completely: ______
$2x^{2} - 8$ completely: ______
$3x^{2} + 6x + 3$ completely: ______
$x^{2} + 7x + 15$ : ______
Three Step-2 branches with one example each:

Two terms: _______________ Example: ______

Three terms: _______________ Example: ______

Four terms: _______________ Example: ______

PAGE 22 — Chapter 13 review, part 1

Chapter 13 Review · Parts A–B

Cells $x^{2}$ , $2x$ , $4x$ , $8$ → factorization ______
$6x^{2} + 15x =$ ______
$-5x^{2} + 20x =$ ______
Explain. Factoring as Chapter 12 backwards:

_______________________________________________

$x^{2} + 11x + 24 =$ ______
$x^{2} - 5x + 6 =$ ______
$x^{2} + x - 12 =$ ______
Is $x^{2} + 7x + 15$ factorable? Justify: _______________________________________________

PAGE 23 — Chapter 13 review, part 2

Chapter 13 Review · Parts C–D

$x^{2} - 9 =$ ______
$4x^{2} - 20x + 25 =$ ______
$2x^{2} + 7x + 3 =$ ______
$6x^{2} - x - 2 =$ ______
$2x^{2} - 8$ completely: ______
$3x^{2} + 6x + 3$ completely: ______
$x^{2} + 8x + 16$ : method ______ factorization ______
Application. Area $x^{2} + 9x + 18$ . Length ______ width ______ Check: ______