SME
Grade 9 Math · Problem type
Binomial times trinomial: the area box
Use this for “Multiply (x + 2)(x² − 3x + 4).”
Diagram
(x + 2)(x² − 3x + 4) = x³ − x² − 2x + 8
Each cell is its row term times its column term; like terms share a diagonal and a tint
This is an example. Type your own values to start a new problem.
(
a
x
+
b
)(
c
x²
+
(
d
)
x
+
e
)
=
p
x³
+
(
q
)
x²
+
(
r
)
x
+
t
A coefficient of 1 is not written: 1x is written x.
Formulas
p
=
a
×
c
1 = 1 × 1
q
=
a
×
d
+
b
×
c
−1 = 1 × (−3) + 2 × 1
r
=
a
×
e
+
b
×
d
−2 = 1 × 4 + 2 × (−3)
t
=
b
×
e
8 = 2 × 4
Assumptions
•
Each term of one factor multiplies each term of the other: 2 × 3 = 6 products.
•
Each cell of the box holds one product: its row term times its column term.
•
Like terms sit on the same diagonal: add them to finish.
Step-by-step
WE KNOW
a = 1, b = 2, c = 1, d = −3, e = 4
FIND
x³ in the product (p), x² in the product (q), x in the product (r), number in the product (t)
Step 1 · Find x³ in the product (p)
Use p = a × c
Only
x
times
x
2
makes
x
3
.
p
=
1
×
1
p
=
1
Step 2 · Find x² in the product (q)
Use q = a × d + b × c
Two
cells
make
x
2
:
x
times
the
x
term,
and
the
number
times
x
2
.
q
=
1
×
(−3)
+
2
×
1
q
=
−1
Step 3 · Find x in the product (r)
Use r = a × e + b × d
Two cells make x: x times the number, and the number times the x term.
r
=
1
×
4
+
2
×
(−3)
r
=
4
−
6
r
=
−2
Step 4 · Find number in the product (t)
Use t = b × e
The two numbers multiply to the number term.
t
=
2
×
4
t
=
8
Check
Put the numbers back in. Both sides should match.
1 = 1 × 1 ✓
−1 = 1 × (−3) + 2 × 1 ✓
−2 = 1 × 4 + 2 × (−3) ✓
8 = 2 × 4 ✓
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Square a binomial
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Factoring polynomials: GCF, trinomials and special products
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