Copy the coordinate plane with the following information Simplify each expression 4 x 5 2x 3 5 x 9 4x 6 6 x 2 2 x 2 2 X fx ID: 436693
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Slide1
Warm-Up
Copy the coordinate plane with the following information.
Simplify each expression.
4) (x + 5) + (2x + 3)
5) (x + 9) – (4x + 6)
6) (-x2 – 2) – (x2 – 2)
X
-
f(x) -
f(x)
X
Slide2
An Intro to Polynomials
Essential Questions: How can we identify, evaluate, add, and subtract polynomials?How can we classify polynomials, and describe their end behavior given the function?Slide3
Classification of a Polynomial by Degree
Degree
Name
Example
-2x
5
+ 3x
4
– x
3
+ 3x
2
– 2x + 6
n = 0
n = 1
n = 2
n = 3
n = 4
n = 5
constant
3
linear
5x + 4
quadratic
2x
2
+ 3x - 2
cubic
5x
3
+ 3x
2
– x + 9
quartic
3x
4
– 2x3 + 8x2 – 6x + 5
quinticSlide4
Classification of a Polynomial by Number of Terms
Name
Example
-2x
5
+
3x
2
–
x
5
-
3x
2
+
6
monomial
3binomial5x + 4trinomial2x2 + 3x - 2polynomial
5x3 + 3x2 – x + 9binomial
Combine like termsSlide5
Example 1a
Classify each polynomial by degree and by number of terms.
a) 5x + 2x3
– 2x2
cubic trinomial
b) x5 – 4x3 – x5 + 3x2 + 4x3
quadratic monomialSlide6
Description of a
Polynomial’s GraphGraph
Example
f(x)= -
2x
5
+ 3x
4
– x
3
+ 3x
2
– 2x + 6
f(x)= 3
f(x)= 5x
+ 4
f(x)= 2x
2
+ 3x - 2f(x)= 5x3 + 3x2 – x + 9f(x)= 3x4
– 2x3 + 8x2 – 6x + 5Slide7
Example 1b
Determine the polynomial’s shape and end behavior.
a)
f(x)= 5x + 2x3 – 2x2
b) f(x)= x5
– 4x3 – x5 + 3x2 + 4x3Slide8
Example 2
Add (5x2
+ 3x + 4) + (3x2 + 5)
= 8x2
+ 3x
+ 9Slide9
Example 3
Add(-3x4
y3 + 6x3y3 – 6x2 + 5xy5
+ 1) + (5x5 – 3x3y3 – 5xy5)
-3x4y
3 + 6x3y3 – 6x2 + 5xy5 + 1
5x5
- 3x3y3 - 5xy5
5x5
– 3x4y3
+ 3x3y3
– 6x2
+ 1Slide10
Example 4
Subtract. (2x2
y2 + 3xy3 – 4y4) - (x2y2
– 5xy3 + 3y – 2y4)
= 2x2y2 + 3xy3 – 4y
4- x2y
2 + 5xy3 – 3y + 2y4
= x2
y2+ 8xy3
– 2y4
– 3ySlide11
Example 4
Subtract.(2a4
b + 5a3b2 – 4a2b3) – (4a4
b + 2a3b2 – 4ab)
2a4b + 5a3b2 – 4a
2b3-4a4
b
-2a
4b- 2a3b2
+ 4ab
+ 3a3b2
– 4a2b3
+ 4abSlide12
Example 5
If the cubic function C(x) = 3x3 – 15x + 15 gives the cost of manufacturing x units (in thousands) of a product, what is the cost to manufacture 10,000 units of the product?
C(x) = 3x
3 – 15x + 15
C(10) = 3(10)3 – 15(10) + 15
C(10) = 3000 – 150 + 15
C(10) = 2865
$2865Slide13
Graphs of Polynomial Functions
Graph each function below.
Function
Degree
# of U-turns in the graph
y = x
2
+ x - 2
2
1
y = 3x
3
– 12x + 4
3
2
y = -2x
3
+ 4x
2
+ x - 2
3
2
y = x
4
+ 5x
3
+ 5x
2 – x - 6
4
3
y = x4 + 2x
3 – 5x2 – 6x
4
3
Make a conjecture about the degree of a function and the # of “U-turns” in the graph.Slide14
Graphs of Polynomial Functions
Graph each function below.
Function
Degree
# of U-turns in the graph
y = x
3
3
0
y = x
3
– 3x
2
+ 3x - 1
3
0
y = x
4
4
1
Now make another conjecture about the degree of a function and the # of “U-turns” in the graph.The number of “U-turns” in a graph is less than or equal to one less than the degree of a polynomial.Slide15
Example 6
Graph each function. Describe its end behavior.
a) P(x) = 2x
3 - 1
b) Q(x) = -3x4 + 2Slide16
Homework