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.
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.