Polynomials
Using the Distributive Property
Polynomial: sum or difference of monomials
Monomial: 1 term (term is constants, variables or
product of constants/variables
Binomial: 2 terms joined by +/Trinomial: 3 terms joined by +/-
Distributive Property:
2a(a2 - 3ab)
-7y(3xy - y)
3m2n(2m2 - 7mn - n2)
-5pr(2pr + p -10r2)
12p2q(-
1
3
q + 2p2q)
2t(4t - 1) - 7(t2 + 3t - 4)
1
Adding and Subtracting Polynomials
Multiply if necessary
Combine like terms
(5x3 - x + 2x2 + 7) + (3x2 + 7 - 4x) + (4x2 - 8 - x3)
Horizontal Method
Vertical Method
(-2x3 - x + 5x2 + 8) - (-2x3 + 3x - 4)
Horizontal Method
Vertical Method
(3x2 - 5x + 3) - 2x(-2x - 3) - 4
FOILING
FOILING
Multiplying 2 binomials
(x - 2)(x + 3)
F (first) O (outer) I (inner) L (last)
x x = x2
x 3 = 3x
x2 +
3x
+
-2 x = -2x -2 3 = -6
-2x
+
-6
x2 + x - 6
(x - 1)(x - 6)
F
F
O
I
(2m + 1)(m - 5)
L
(3t + 5)(t + 3)
O
I
L
F
F
O
I
L
(y - 4)(y + 4)
O
I
L
2
Special Products
Crossing 2 pink snapdragons
Pink snapdragons are half red, half white (.5R + .5W)
Crossing 2 pink = (.5R + .5W)(.5R + .5W) or = (.5R + .5W)2
R
W
(.5R + .5W)(.5R + .5W)
R
W
Perfect Square Trinomials:
(a + b)2
Try: (x + 2)2
(2m + 1)2
(a - b)2
Try: (x - 3)2
(2m - 1)2
Difference of perfect squares:
(a + b)(a - b)
Try: (x + 2)(x - 2)
(2m + 1)(2m - 1)
(x + 5)(x - 5)
(n - 3)(n + 3)
3
Zero Product Property
if ab = 0, then a =0, b = 0 or a & b = 0
Examples:
(x - 2)(x + 3) = 0
(2x + 1)(3x - 2) = 0
(x - 4)(x + 5) = 0
When y = 0, you get the x-intercepts of a trinomial,
a.k.a._______________________ or ______________________
Find the vertex the old-fashioned way, and graph, using the vertex and
roots
y = (x - 4)(x + 1)
y = (x - 4)(x + 4)
4
1. y = (2x - 4)(x - 2)
3. y = (2x - 2)(x + 4)
5. y = 2x2 + 16x + 30
2. y = 2x2 - 6x - 8
4. y = (-x - 3)(x - 4)
6. y = (-x - 6)(x - 6)
5
Factoring
Factoring Out Monomials
"undistribute" the GCF
makes factoring trinomials easier
5n - n2
24x + 48y
12x3y2 - 18x2y2 + 30x
Factoring Trinomials
Find the numbers that add to equal "b"
and multiply to equal "c"
factors
x2 + 7x + 12
m2 - 14m + 40
y2 - 5y - 24
-a2 + 16a + 36
Factoring with a leading coefficient (trial and error):
3x2 + 2x - 8
7a2 + 22a + 3
2x2 - 5x - 12
3c2 - 3c - 5
4n2 - 4n - 35
6m2 + 19mn + 10m2
-4y2 + 19y - 21
Factoring the difference of perfect squares
n2 - 25
4y2 - 169
8m2 - 242
196x2 - 16y2
6
7
INVERSE VARIATION
Remember Direct Variation
- graph was a ________________
- as x _____________, y _________________
- equation looked like: ________________
- "k" was the ______________ ___ _____________
Inverse variation: As x increases, y decreases (like negative correlation)
at a constant rate, "k"
the inv variation equation is y =
k
x
the formula for k = xy
Example:
When you walk, time is inversely related to speed
(if you take
a 5mile walk, the faster you walk, the less time it takes)
Area covered by a truckload of mulch and the depth of the mulch
Time it takes to paint a house and the # of painters
(2 painters take 6 hours
3 painters take 4 hours
4 painters take 3 hours...)
The equation for the relationship is: y =
12
x
Where y = # of hours and x = # painters
(the constant of variation (k) = xy or 6hrs x 2 painters
4 hrs x 3 painters
3 hrs x 4 painters
find the equation if x & y vary inversely:
x = 2, y = 5
x = 13/5, y = 5
x = 3, y = 7
If the volume of a prism (vol = Bh)= 10m3, the base and height are inversely
related.
What is "k"?
What is the equation that relates x to y?
What is the measure of the base when the height = 4m?
8
Graphing Quadratics by Factoring
1
2
3
Find the vertex
Set the trinomial equal to zero to find the x-intercepts
factor the trinomial into 2 linear factors
use zero product property to solve for zeros
Graph the vertex and intercepts/zeros
y = 10x2 - 7x - 12
y = x2 - 3x - 18
y = x2 - 6x + 9
Spe
cial
Cas
e!
9