Multiplying PolynomialsMade Easy

Punnett Squares are often used in Biology to help explain Genetics. We can use them in Algebra to help explain multiplying polynomials.(x +

3) (x – 5)

(x – 2) (x + 2)

(x + 1) (x2 + 3x + 4)

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.

(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.

(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

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

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

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?

The first step is to make sure the trinomial is in descending order.

X2 + 5x + 6

Second degree First

degreeConstant

This trinomial is already in the proper order.

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?

15 6

Is there a number common to all three that can be divided out? Not on this particular polynomial.

X2 + 5x + 6

We’re ready to begin!x2

6

The x2 te

rm always g

oes here.

The consta

nt always g

oes here.

Now we’re ready to figure out the middle terms.

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.

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 = 62 * 3 = 6

1 + 6 = 7

2 + 3 = 5

There’s our answer! 2x and 3x will fill our Punnett Square.

2x

3x

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)

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)

Finally, it’s time to put it all together! The trinomial

x2

62x

3x

x +3

x

+2X2 + 5x + 6Factors as

(x + 3) (x + 2)

Yay! We did it!

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!

The first step is to make sure the trinomial is in descending order.

8x2 - 2x - 3

Second degree First

degreeConstant

This trinomial is already in the proper order.

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?

82 3

Is there a number common to all three that can be divided out? Not on this particular polynomial.

8x2 - 2x - 3

We’re ready to begin!8x2

-3

The x2 te

rm always g

oes here.

The consta

nt always g

oes here.

Now we’re ready to figure out the middle terms.

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.

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 = 242 * 12 = 243 * 8 = 244 * 6 = 24

1 - 24 = -23 24 – 1 = 232 – 12 = -10 12 – 2 = 103 – 8 = -5 8 – 3 = -54 - 6 = -2 6 – 4 = 2

There’s our answer! -6x and 4x will fill our Punnett Square.

4x

-6x

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)

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)

Finally, it’s time to put it all together! The trinomial

8x2

-34x

-6x

4x -3

x

+18x2 - 2x - 3Factors as

(4x - 3) (2x + 1)

Yay! We did it!

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!