Degree and Lead Coefficient
End Behavior

The polynomial is not in the correct order
3x 3 + 2 – x 5 + 7x 2 + x
Just move the terms around
-x 5 + 3x 3 + 7x 2 + x + 2
Now it is in correct form

When the polynomial is in the correct order
Finding the lead coefficient is the number in front of the first term
-x 5 + 3x 3 + 7x 2 + x + 2
Lead coefficient is – 1
It degree is the highest degree
Degree 5
Since it only has one variable, it is a
Polynomial in One Variable

Evaluate a Polynomial
To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1 Let x = 4, 5, and 6

Evaluate a Polynomial
To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4) 2 – 3(4) + 1 = 37
= 3(16) – 12 + 1
= 48 – 12 + 1
= 36 + 1 = 37

Evaluate a Polynomial
To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4) 2 – 3(4) + 1 = 37 f(5) = 3(5) 2 – 3(5) + 1 =

Evaluate a Polynomial
To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4) 2 – 3(4) + 1 = 37 f(5) = 3(5) 2 – 3(5) + 1 = 61
= 3(25) – 15 + 1
= 75 – 15 + 1 = 61

Evaluate a Polynomial
To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4) 2 – 3(4) + 1 = 37 f(5) = 3(5) 2 – 3(5) + 1 = 61 f(6) = 3(6) 2 – 3(6) + 1 =

Evaluate a Polynomial
To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4) 2 – 3(4) + 1 = 37 f(5) = 3(5) 2 – 3(5) + 1 = 61 f(6) = 3(6) 2 – 3(6) + 1 = 91
= 3(36) – 18 + 1 = 91

Find p(y 3 ) if p(x) = 2x 4 – x 3 + 3x

Find p(y 3 ) if p(x) = 2x 4 – x 3 + 3x p(y 3 ) = 2(y 3 ) 4 – (y 3 ) 3 + 3(y 3 ) p(y 3 ) = 2y 12 – y 9 + 3y 3

Find b(2x – 1) – 3b(x) if b(m) = 2m 2 + m - 1
Do this problem in two parts b(2x – 1) =

Find b(2x – 1) – 3b(x) if b(m) = 2m 2 + m - 1
Do this problem in two parts b(2x – 1) = 2(2x – 1) 2 + (2x -1) – 1
=2(2x – 1)(2x – 1) + (2x – 1) – 1
=2(4x 2 – 2x -2x + 1) + (2x -1) – 1
= 2(4x 2 – 4x + 1) + (2x – 1) -1
= 8x 2 – 8x + 2 + 2x -1 – 1
= 8x 2 - 6x

Find b(2x – 1) – 3b(x) if b(m) = 2m 2 + m - 1
Do this problem in two parts b(2x – 1) = 8x 2 - 6x
-3b(x) = -3(2x 2 + x – 1) = -6x 2 – 3x + 3 b(2x – 1) – 3b(x) = (8x 2 – 6x) + (-6x 2 – 3x + 3)
= 2x 2 – 9x + 3

End Behavior
We understand the end behavior of a quadratic equation. y = ax 2 + bx + c both sides go up if a> 0 both sides go down a < 0
If the degree is an even number it will always be the same.
y = 6x 8 – 5x 3 + 2x – 5 go up since 6>0 and 8 the degree is even

End Behavior
If the degree is an odd number it will always be in different directions.
y = 6x 7 – 5x 3 + 2x – 5
Since 6>0 and 7 the degree is odd raises up as x goes to positive infinite and falls down as x goes to negative infinite.

End Behavior
If the degree is an odd number it will always be in different directions. y = -6x 7 – 5x 3 + 2x – 5
Since -6<0 and 7 the degree is odd falls down as x goes to positive infinite and raises up as x goes to negative infinite.

End Behavior
If a is positive and degree is even, then the polynomial raises up on both ends
(smiles)
If a is negative and degree is even, then the polynomial falls on both ends
(frowns)

End Behavior
If a is positive and degree is odd, then the polynomial raises up as x becomes larger, and falls as x becomes smaller
If a is negative and degree is odd, then the polynomial falls as x becomes larger, and rasies as x becomes smaller

Tell me if a is positive or negative and if the degree is even or odd

Tell me if a is positive or negative and if the degree is even or odd a is positive and the degree is odd

Tell me if a is positive or negative and if the degree is even or odd

Tell me if a is positive or negative and if the degree is even or odd a is positive and the degree is even

Tell me if a is positive or negative and if the degree is even or odd

Tell me if a is positive or negative and if the degree is even or odd a is negative and the degree is odd

Page 350 – 351
# 17 – 27 odd, 31,
34, 37, 39 – 43 odd

Homework
Page 350 – 351
# 16 – 28 even, 30,
35, 40 – 44 even