Light can travel 186,000 miles in one minute. In scientific notation, this is 1.86x105 miles. The document discusses multiplying polynomials and binomials. It introduces the box method and the F.O.I.L (First, Outer, Inner, Last) method for multiplying binomials. F.O.I.L involves distributing the first term of one binomial with each term of the other binomial. The sign of the middle and last terms depends on whether the signs of the last terms of each binomial are the same or different.

2. Warm-Up:
The speed of light is 186,000 miles per second. How far
can light travel in one minute? Write your answer in
scientific notation.

3. Review From Yesterday:
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:
(xz + 5x²z – x) + (x+ 5z)
(x + z) + (zx+ z²x)
(a + 5ba) + (3ba+ a²) (3xyz - xyz + zx) + (3zyx+ 1)
Like terms have the same exponent to the same degree.......
But order does not matter!

4. Multiplying Polynomials:
Let's begin by multiplying a monomial by a monomial.
= 2x7
If the bases (x) are the same,
we add the exponents
2x7y
(2x ) • (yx ) =
Now multiply a monomial by a binomial
3
4
Once more: x(7x2 + 4y) = 7x3 + 4xy
7x3 + 4x
When multiplying polynomials, each term is multiplied by
every other term.
Now we look at multiplying a binomial by a binomial.

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

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

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

9. 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

10. Multiplying Binomials Mentally
Find the 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 + 9x + 20
x2 + 4x + 5x + 20
(x + 6)(x + 5)
x2 + 5x + 6x + 30
x2 + 11x + 30
There are lots of patterns here, but this one
enables us to multiply
(x + a)(x + b) = x2 + x(a + b) + ab
binomials mentally.
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.

11. 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

12. 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

13. 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 larger doesn't matter when both signs
are the same, but it does when the signs are different.

14. Remember, F.O.I.L can be used when
multiplying a binomial by another binomial.

15. Class Work:
See Handout