Nutcracker NickTilewrights · polynomials and factoring

Cracks a trinomial into two binomials: factoring x squared plus bx plus c.

Factoring Trinomials (x² + bx + c)

Factoring x squared plus bx plus c. In MathBound, the free Algebra I adventure game, Nutcracker Nick asks these problems, and the only way past is to solve them.

Part of Polynomials and Factoring: chapter 8 of the standard course.

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10 questions, no sign-up, nothing saved. Climbs from the basics to harder questions.

What you will practice

Positive Constant. Factor a trinomial where the constant and middle terms are positive (requires two positive factors).
Negative Constant. Factor a trinomial where the constant is negative and the middle term is positive (larger factor is positive).
Two Variables. Factor a trinomial of the form x^2 + bxy + cy^2 containing two variables.
Factor GCF First. Factor out a positive integer GCF before factoring the resulting trinomial.

Worked examples

Made by the game’s own question engine. Try each one, then open its steps.

Example 1 · Positive Constant

A

rectangle

has

area

x^{2} + 11 x + 30

and

width

(x + 5) .

Which

expression

gives

its

length?

$(x + 5)$
$(x + 7)$
$(x - 6)$
$(x + 6)$

Show the worked steps

$Area$ $is$ $length$ $times$ $width,$ $so$ $factor$ $the$ $area:$ $(x + 6) (x + 5) .$
$Without$ $the$ $width$ $(x + 5),$ $the$ $length$ $is$ $(x + 6) .$

Answer: $(x + 6)$

Example 2 · Negative Constant

For

x^{2} + x - 42,

the

numbers

- 6

and

7

multiply

to

- 42

and

add

to

1 .

Which

factoring

do

they

give?

$(x + 1) (x - 42)$
$(x - 7) (x + 6)$
$(x + 7) (x + 6)$
$(x + 7) (x - 6)$

Show the worked steps

$- 6 \cdot 7 = - 42$ $and$ $- 6 + 7 = 1 .$
$They$ $are$ $the$ $constants$ $of$ $the$ $brackets:$ $(x + 7) (x - 6) .$

Answer: $(x + 7) (x - 6)$

Example 3 · Two Variables

Factor:

x^{2} - 10 x y + 24 y^{2}

$(x - 6 y) (x - 4 y)$
$(x - 6 y) (x + 4 y)$
$(x - 10 y) (x + 24 y)$
$(x + 6 y) (x + 4 y)$

Show the worked steps

$Treat$ $y$ $as$ $part$ $of$ $the$ $constant:$ $find$ $two$ $numbers$ $multiplying$ $to$ $24$ $and$ $adding$ $to$ $- 10 .$
$- 6$ $and$ $- 4$ $work.$
$So$ $the$ $factors$ $are$ $(x - 6 y) (x - 4 y) .$

Answer: $(x - 6 y) (x - 4 y)$

Where it is in your textbook

Textbook	Lessons
Standard course	8.5
Big Ideas Math Algebra 1	7.5
Big Ideas Math Algebra 1 (2022)	7.5
Carnegie Learning Algebra I	M5 2.4
CPM Core Connections Algebra	8.1.2, 8.1.4
CPS Skyline Algebra	7.6-7.7
enVision Algebra 1	7-5
enVision Algebra 1 (2024)	7-5
Eureka Math Algebra I (EngageNY)	4.1-4.2
Glencoe Algebra 1	8-6
HMH Into Algebra 1	17.2
Holt McDougal Larson Algebra 1	8.5
Illustrative Mathematics Algebra 1	7.6-7.8
Illustrative Mathematics Algebra 1 (v.360)	8.6-8.8
Math Nation Algebra 1 (Florida B.E.S.T.)	7.2, 7.3, 7.5
Open Up NC Math 1 (MVP)	7.3, 7.4
OpenStax Elementary Algebra 2e	7.2
Pearson Algebra 1 Common Core	8-5
Reveal Algebra 1	10-6
SpringBoard Algebra 1	27

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