Today's math lesson covers dividing polynomials by monomials and multiplying binomials. Students should bring good notes and their full attention. The warm-up reviews solving binomial equations and discount word problems. Methods for multiplying binomials include the box method and FOIL (First, Outer, Inner, Last). Patterns emerge depending on the signs of terms. Dividing polynomials begins with dividing each term in the numerator by the monomial denominator. Students should take complete, easy to read notes on multiplying and dividing polynomials.

1. Today's Goals:
Dividing Polynomials by a Monomial
 Multiplying Two Binomials
What You'll Need:
 Good Notes
Your Full Attention
 Review New Khan Academy Topics

2. Warm-Up:
2. The sum of 2 binomials is 5x2
- 6x. If
one of the binomials is 3x2
- 2x, what is
the other binomial?
1. 5(4x - 3) - 3 = 37
2
4. Solve in 1 Step: What is the sale price of a $70 pair of shoes
discounted 20%?

3. This Week's Khan
Tomorrow:

4. Review From Last Week:
Like terms have the same exponent to the same degree.
When adding or subtracting, only like terms can be
combined. When polynomials have more than one variable,
the same rules apply. For example, Simplify:
(xz + 5x²z – x) + (x+ 5z) (x + z) + (zx+ z²x)
(a + 5ba) + (3ba+ a²)
Like terms have the same exponent to the same degree.......
But order does not matter!

5. Multiplying Polynomials:
Let's begin by multiplying a monomial by a monomial.
= 2x7
(2x3
) • (yx4
)
=
If the bases (x) are the same,
we add the exponents
2x
7
y
Now multiply a monomial by a binomial
Once more: x(7x
2
+ 4y) = 7x
3
+ 4x7x
3
+ 4xy
When multiplying polynomials, each term is multiplied by
every other term.
Now we look at multiplying a binomial by a binomial.

6. Method #1: The Box Method
Multiplying Binomials
(x + 4)(x + 2)
*Reminder: When multiplying,
add the exponents if bases are alike
Binomials

7. Multiplying Binomials
=
=
Use the Box Method:
x2
-3x
+4x -12

8. Multiplying Binomials
=
However, the More Common Method
for solving binomials is...
=

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

11. Multiplying Binomials Mentally
(x + 2)(x + 1)
(x + 3)(x + 2)
(x + 4)(x + 3)
(x + 5)(x + 4)
(x + 6)(x + 5)
x2 + x + 2x + 2
x2 + 2x + 3x + 6
x2 + 3x + 4x + 12
x2 + 4x + 5x + 20
x2 + 5x + 6x + 30 x2 + 11x + 30
x2 + 9x + 20
x2 + 7x + 12
x2 + 5x + 6
x2 + 3x + 2
The middle term of the answer is the sum of the binomial's last
terms and the last term in the answer is the product of the
binomial's last terms.
(x + a)(x + b) = x2 + x(a + b) + ab
There are lots of patterns, but this one enables us to multiply
binomials mentally.
Find the pattern

12. 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:
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.
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
(x + 4)(x - 3) = x2 + x - 12
(x - 4)(x + 3)

13. Two Negatives
What happens if the second term in both binomials is negative?
Example:
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)
(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

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

16. Remember, F.O.I.L can be used when
multiplying a binomial by another binomial.
It is one method, but not the only method for multiplying binomials.
•
•
•

17. Dividing Polynomials
Dividing a Polynomial by a binomial
Please take complete, easy to
read notes, you will need them.

18. Dividing Polynomials
Part I: Dividing by a Monomial:
1. Since the denominator is the same for each term, divide each
term in the numerator by the denominator. The result is now a
monomial divided by a monomial for each term.
Solve.
2x2
18x4
-10x2
+ 6x7
9x2
-5 + 3x5

19. Class Work:
See Handout