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Warm up
Simplify
1) a + 4b - 3a - b
4) (-4a2b)(3ab3)
2) 2(x - 3) - (4x - 5)
3) 3x(4x - 7) + x(x + 5)
5) 20x3y4
-5x2y2
Unit 2: Polynomials
Polynomial - algebraic expression formed by adding or subtracting terms
Term - the product of a number and possibly one or more variables.
Examples: Term
Coefficient
Variable
Degree of a Term - sum the exponents of the variables
Example:
Degree of a Polynomial - the degree of the highest degree term
Example:
Complete the Table
Polynomial
Number
of Terms
6
Classified by
Number of Terms
Degree
Classified
by Degree
3x
4x + 2
x2 + 4x + 1
3x3 + x2 - x - 7
Practice: p. 131: #1, 2, 3, 7, 8, 12, 16
Warm up
Simplify
1) 2x(3x - 4) - (x + 1)
2) (2a2b)(-3ab2)
Multiplying Binomials
Examples: Expand and Simplify
1) (2x + 4)(3x + 1)
2) (x + 1)(x + 2)
3) (3a + b)(a - 4b)
4) 3(2x - 1)(x - 4)
5) 2(3y + 2)(y - 1) - (y - 2)(2y + 1)
p. 137: #1 – 5 (column 1), 8, 12
3) (4x3y)2
2y2
Warm up
Expand and Simplify
1) (m - 1)(m - 7)
2) 2(3j - 1)(4j +2)
3) 2x(3x + 4) - (6x + 5)(x - 1)
Special Products
Examples: Expand and Simplify
1) (3x + 1)2
2) (a - 3)2
Squaring binomials results in a perfect square trinomial.
In general: (a + b)2 =
(a - b)2 =
Expand and Simplify
3) (2x + 5)(2x - 5)
Multiplying identical binomials, except for the sign, results in a difference of squares.
In general, (a - b)(a + b) =
4) Simplify
2(3x + 4)2 - (4x + 5)(4x - 5)
p. 142: #3, 4 - 8, 18 (a, c, e, ... for all of them), 20
Common Factoring
To factor an expression is to write it as a product
Example 1: Factor by factoring out the greatest common factor
a) 3a - 9
b) x2 + 2x
c) 5a2 + 10a
d) 2x3 + 6x2 - 12x
e) 10x3 - 15x2
f) 3x2y + 12xy
g) 15a3b4c + 20a2b5c3
Example 2: Factors can be polynomials sometimes
1) a(2x+1)+b(2x+1)
2) 2(a+b)-3c(a+b)
Example 3: Factor by grouping
1) bx+3x+by+3y
p. 150 #1 - 4 (every other part), 5, 7
2) 9m +12 - 15m2-20m
Warm up
Factor by factoring out the greatest common factor.
1) 5b - 10
2) -8a + 12
3) 14x4 - 21x3
4) 6a5b4 - 12a3b3 + 18a4b2
Factoring Trinomials: x2 + bx + c
(x + 4)(x + 2) =
Steps for factoring trinomials of the form x2 + bx + c
1) Write two brackets with x at the front of each.
2) Fill in two numbers that 3) Check by expanding.
Examples: Factor
x2 - 4x + 3
x2 + 14x + 40
x2 - 7x + 12
a2 - 4a - 21
-3n2 + 3n + 90
x2 + 2xy - 48y2
p. 156 #2 - 6 (first column), 8, 10, 11, 12
Warm Up
Factor
a) x2 - x - 6
b) x2 + x – 6
c) 2x2 - 18x + 40
Factoring Harder Trinomials
Factor completely
4x2 - 8x - 12
common factor
sum & product
Not so tricky... but! Factor 2n2 + 7n + 6
1) Factor 2n2 + 7n + 6 using a chart
2) Factor 2n2 + 7n + 6 using decomposition
Examples: Factor
a) 3a2 - 17a + 20
b) 6p2 + 11p – 10
c) 8n2 - 13n – 6
d) 16n2 - 26n - 12
Question:
2x - 5 is a factor of 2x2 + 9x - 35. What is the other factor?
How can you check to see if you have factored correctly?
p. 163 #1 - 4 (first column)
Warm up
Factor
1) -3x2 - 12x
2) 2x2 - 8x - 42
3) 5x2 + x - 6
Harder Trinomials Continued
Factor
1) 8x2 -10xy - 3y2
2) 10r4 - 22r2 + 4
3) -20g2 - 34g - 6
Today's Practice Problems
p. 163 # 4 (middle column), 5, 8 ,9
Warm up
Factor
1) x2 - 6x + 9
2) 4x2 + 12x + 9
Perfect Square Trinomials
Factor
1) t2 -12t + 36
2) 25y2 + 40yz + 16z2
Pattern: a2x2 + 2abx + b2 = (ax + b)2
a2x2 - 2abx + b2 = (ax - b)2
Difference of Squares
Factor
x2 - 25
Pattern: a2x2 - b2 = (ax + b)(ax - b)
Factor
1) 49y2 - 36
2) 36 - 9k2
4) a4 – 16
5) (x + 3)2 - 25
p. 167 #1 - 3 (first column), 6 - 7 (a, c, e...), 8
3) 28x2 - 175y2