The document discusses various methods for multiplying binomial expressions, including the distributive property, the box method, and the F.O.I.L. (First, Outer, Inner, Last) method. It also covers patterns that emerge when multiplying binomials mentally and how the signs of terms impact the results. Practice problems are provided to help solidify these skills.

1. Today
:Warm Up
Final exam
review
Binomials
polynomials
Classwork
test friday on
exponents and
scientific notation

2. Warm- Up Exercises
1. Find the total area of the figure.
2. What is the solution to:
-4y -20 = -10x and -5x - 14 = -2y
4. Write any expression in
which a monomial is
5. Simplify:
multiplied by a binomial.
3m2(3m + 2n - 4p)
3. List 3 different types of
Monomials.

3. Multiplying Binomials
#1: The Box Method (x + 4)(x + 2)
*Reminder: When
multiplying, add the exponents

4. Multiplying Binomials
=
=
The More Common Method for
solving binomials is...

6. Multiplying Binomials
We know how to multiply a binomial by a monomial:
a ( x + 2) = ax + 2a
Can we use the distributive property to multiply a binomial by a binomial?
Suppose a = (x + 1). How do we find this product:
(x + 1) ( x + 2) ?
Can we distribute (x + 1) across (x + 2) ? The answer is yes.
First multiply (x + 1) ( x ).
Then multiply (x + 1) ( 2 ) .
(x + 1) (x + 2) = (x + 1) ( x ) + (x + 1) ( 2 )
(x2 + x) + (2x + 2)
x2 + 3x + 2

7. F.O.I.L.
If we perform our distribution in this order,
(x + 1)(x + 2) = x (x + 2) + 1 (x + 2)
a useful pattern emerges.
Distributing produces the sum of these four multiplications.
First + Outer + Inner + Last
"F.O.I.L" for short.
(x + 1) (x + 2) = x ( x + 2 ) + 1 ( x + 2 ) x2 + 2x + x + 2
x2 + 3x + 2

8. Multiplying Binomials Mentally
Can you see a pattern?
(x + 2)(x + 1) x2 + x + 2x + 2 x2 + 3x + 2
(x + 3)(x + 2) x2 + 2x + 3x + 6 x2 + 5x + 6
(x + 4)(x + 3) x2 + 3x + 4x + 12 x2 + 7x + 12
(x + 5)(x + 4) x2 + 4x + 5x + 20 x2 + 9x + 20
(x + 6)(x + 5) x2 + 5x + 6x + 30 x2 + 11x + 30
There are lots of patterns here, but this one
(x + a)(x + b) = x2 + (a + b) x + ab
enables us to multiply binomials mentally.
Later we will use this pattern "in reverse" to factor
trinomials that are the product of two binomials.

9. Practice: Multiplying Binomials Mentally
1. What is the last term when (x + 3) is multiplied by (x + 6) ?
18 18 = 6 times 3
2. What is the middle term when (x + 5) is multiplied by (x + 7) ?
12x 12 = 5 plus 7
3. Multiply: (x + 4) (x + 7)
x2 + 11x + 28 4 plus 7 = 11 4 times 7 = 28
4. Multiply: (x + 7) (x + 4)
x2 + 11x + 28 7 plus 4 = 11 7 times 4 = 28

10. Positive and Negative
All of the binomials we have multiplied so far have been sums of
positive numbers. What happens if one of the terms is negative?
Example 1: (x + 4)(x - 3)
1. The last term will be negative, because a positive
times a negative is negative.
2. The middle term in this example will be positive,
because 4 + (- 3) = 1.
(x + 4)(x - 3) = x2 + x - 12
Example 2: (x - 4)(x + 3)
1. The last term will still be negative, because a positive
times a negative is negative.
2. But the middle term in this example will be negative,
because (- 4) + 3 = - 1.
(x - 4)(x + 3) = x2 - x - 12

11. Two Negatives
What happens if the second term in both binomials is negative?
Example: (x - 4)(x - 3)
1. The last term will be positive, because a negative
times a negative is positive.
2. The middle term will be negative, because a negative
plus a negative is negative.
(x - 4)(x - 3) = x2 -7x +12
Compare this result to what happens when both terms are positive:
(x + 4)(x + 3) = x2 +7x +12
Both signs the same: last term positive
middle term the same

12. Sign Summary
Middle Term Last Term
(x + 4)(x + 3) positive positive
(x - 4)(x + 3) negative negative
(x + 4)(x - 3) positive negative
(x - 4)(x - 3) negative positive
Which term is bigger doesn't matter when both signs
are the same, but it does when the signs are different.

13. Remember, F.O.I.L can be used when
multiplying any binomial by another binomial.

14. Class Work:
Handout on Multiplying Binomials