How to use Punnett Squares to multiply and factor polynomials

1. Multiplying Polynomials
Made Easy
Punnett Squares are often used in Biology to
help explain Genetics. We can use them in
Algebra to help explain multiplying polynomials.

2. What are “Punnett Squares”?
They are nothing more than large grids.
You can use them to input any information
that needs to be multiplied together.
Let’s have some
fun seeing how
Punnett
Squares can
help us learn
Algebra.

3. (x + 3) (x + 2)
x
x
+2
x2
2x
3x
6
Answer: x2 + 2x + 3x + 6 = x2 + 5x + 6
+3Let’s start with the
following problem.
Multiply:
Multiply row by column.

4. (x + 2) (x2 + 2x + 3)
x +2
x2
+2x
+3
x3
2x2
3x
2x2
4x
6
Answer: x3 + 4x2 + 7x + 6
Let’s try another
problem. Multiply
row by column

5. Now that we have seen how
Punnett Squares can help us
learn to multiply polynomials,
let’s get to work!

6. Punnett Squares
Part 2
Using them to
factor trinomials
We can use Punnett Squares to factor
trinomials very easily. Let’s take a trip
down Algebra Road to success!
x2
2x
3x
6

7. Now that we know
how to use Punnett
Squares to multiply
polynomials, let’s see
if we can use them to
factor trinomials.
X2 + 5x + 6
X2
62x
3x
???????
How did that happen?

8. The first step is to
make sure the
trinomial is in
descending order.
X2 + 5x + 6
Second
degree First
degree
Constant
This trinomial
is already in
the proper
order.

9. X2 + 5x + 6
The next step is to check
for a common factor that
can be factored out. What
are the coefficents of each
term?
1
5
6
Is there a number
common to all three
that can be divided
out? Not on this
particular polynomial.

10. X2 + 5x + 6
We’re ready to begin!
x2
6
Now we’re ready
to figure out the
middle terms.

11. First we multiply the
coefficient of the x2
term and the constant
together.
X2 + 5x + 6
1(6) = 6
x2
6
This sign tells us
the factors are
the same sign
This sign
tells us both
factors are
positive
Now we check the
signs, looking at the
second sign first,
then at the first sign.

12. x2 + 5x + 6
Now we need to determine
the factors of 6 because we
will add them together to
equal the x term.
x2
6
1 * 6 = 6
2 * 3 = 6
1 + 6 = 7
2 + 3 = 5
There’s our answer! 2x and 3x
will fill our Punnett Square.
2x
3x

13. x2
62x
3x
First let’s look at
columns for common
factors. Remember,
the sign on the x term
is the sign for that
factor.
x +3
In the first column, x is common
to both terms. In the second
column, positive 3 is common to
both terms.
We have found
our first factor!
It is
(x + 3)

14. x2
2x
3x
6
x +3Now let’s look at rows.
Remember, the sign on
the x term is the sign
for that factor
In the first row, x is common to
both terms. In the second row,
positive 2 is common to both
terms.
x
+2
We have found our
second factor! It is
(x + 2)

15. Finally, it’s time to
put it all together!
The trinomial x2
62x
3x
x +3
x
+2
X2 + 5x + 6
Factors as
(x + 3) (x + 2)

16. Factoring Trinomials
with a
Leading Coefficient
Other Than 1
We can use Punnett Squares to
factor trinomials with a leading
coefficient other than one. It’s
really easy!

17. The first step is to
make sure the
trinomial is in
descending order.
8x2 - 2x - 3
Second
degree First
degree
Constant
This trinomial
is already in
the proper
order.

18. 8x2 - 2x - 3
The next step is to check
for a common factor that
can be factored out. What
are the coefficents of each
term?
8
2
3
Is there a number
common to all three
that can be divided
out? Not on this
particular polynomial.

19. 8x2 - 2x - 3
We’re ready to begin!
8x2
-3
Now we’re ready
to figure out the
middle terms.

20. First we multiply the
coefficient of the x2
term and the constant
together.
8x2 - 2x - 3
8(3) = 24
8x2
-3
This sign tells us
the factors are
different signs
This sign
tells us the
larger factor
is negative
Now we check the
signs, looking at the
second sign first,
then at the first sign.

21. 8x2 -2x - 3
Now we need to determine
the factors of 24 because we
will find the difference to
equal the x term.
8x2
-3
1 * 24 = 24
2 * 12 = 24
3 * 8 = 24
4 * 6 = 24
1 - 24 = -23 24 – 1 = 23
2 – 12 = -10 12 – 2 = 10
3 – 8 = -5 8 – 3 = -5
4 - 6 = -2 6 – 4 = 2
There’s our answer! -6x and
4x will fill our Punnett Square.
4x
-6x

22. 8x2
-34x
-6x
First let’s look at
columns for common
factors. Remember,
the sign on the x term
is the sign for that
factor.
4x -3
In the first column, 4x is
common to both terms. In the
second column, negative 3 is
common to both terms.
We have found
our first factor!
It is
(4x - 3)

23. 8x2
4x
-6x
-3
4x -3Now let’s look at rows.
Remember, the sign on
the x term is the sign
for that factor
In the first row, 2x is common
to both terms. In the second
row, positive 1 is common to
both terms.
2x
+1
We have found our
second factor! It is
(2x + 1)

24. Finally, it’s time to
put it all together!
The trinomial 8x2
-34x
-6x
4x -3
x
+1
8x2 - 2x - 3
Factors as
(4x - 3) (2x + 1)

25. Remember, Punnett Squares are an easy way to
multiply and factor polynomials, but they are not the
only way. If you have already learned to do these
tasks with other methods and can use those
methods successfully, you may prefer to stick with
the “tried and true”. Even if you prefer to do that,
give Punnett Squares a chance. They might just
make the job easier!
Success!