Unit 5 Notes 5A I can apply all operations on monomials and polynomials. Properties of Exponents: xm • xn = ________________ (xm)n = _________________ (xy)m = ___________________ x-­‐m = ____________________ !
! !!
=___________________ !!
x0 = __________________ ! ! =___________________ Simplify each expression: 1) (4x2 – 5x + 6) – (3x – 1)
! !!
!
! !
!
2) (6x4 – 7x + 8) + (-4x3 + 9x -5)
−2
4) (2a – b)(4a + b)(3b – 5)
5)
= ___________________ _________________ 3) (x – 4) (x2 + 4x + 16)
3
4x y
5 −6
32x y
6) (𝑥 ! + 𝑦 ! )(𝑥 ! − 𝑦 ! ) 4x y
9) 5 −6
32x y
3
−2
3 2
−2
7). (2a )(3a b )(c ) ⎛ −2a 4 ⎞
⎜ b 2 ⎟⎠
8) ⎝
−2
3
10) Simplify 6x
4 y 3 +
12x 3 y 2 − 18x
2 y 11) (18x2y + 27x3y2z) (3xy)-­‐1 3xy
5B I can divide polynomials using long division and synthetic division. Ex. 1: Use long division to find each quotient: 2x 3 + 6x + 152
x+4
B) A) (x + 3x − 40) ÷ (x − 3) Example 2: Use Synthetic division to find each quotient: 3
2
4
3
A) 2x − 13x + 26x − 24 ÷ ( x − 4 ) B) 6b − 8b + 12b − 14 ÷ ( b − 2 )
2
(
)
(
)
5C I can describe features of a polynomial graph. Degree of Polynomial: highest exponent The degree of a polynomial tells how many ROOTS/SOLUTIONS the function has. Leading Coefficient: The coefficient of the term with the highest exponent. The degree of a polynomial has important implications for how the graph of the function will look
Create a graph of a polynomial function with the given constraints
(Consider the End Behavior, and how many roots the equation could have)
-10
8
8
6
6
4
4
2
2
-5
5
10
-10
-5
5
-2
-2
-4
-4
-6
-6
-8
-8
10
Relative Maximum/Minimum:
Turning Point: Any place where the graph curves. Maximum # of Turning Points= Degree-1. (If degree is
5 the graph could have at most 4 turning points.
Graph a polynomial with degree
6, relative Max of 4, relative
Min of -2, and 4 real zeros Graph a polynomial with degree 5,
relative Max of 5, relative Min of 3,
and 1 real zero 8
4
4
2
2
2
5
-2
-4
-6
-8
Describe the end behavior of the
graph: 6
6
4
-5
8
8
6
-10
Graph a polynomial with degree 7,
relative Max of 2, relative Min of
-6, and 5 real zeros 10
-10
-5
5
-2
-4
-6
-8
Describe the end behavior of the
graph: 10
-10
-5
5
-2
-4
-6
-8
Describe the end behavior of the
graph: 10
5D I can solve polynomial equations by factoring.
Factoring Techniques-LEARN THESE!!!!!!!
# of Terms
Factoring Technique
GCF – What can I factor
Any Number
out of each term?
2
3
General Case
Example
2𝑥 ! + 3𝑥 ! =𝑥 ! (2+3x)
8𝑥 ! + 4𝑥 ! − 2𝑥 ! =
2𝑥 ! (4𝑥 ! +2x-1)
Difference of Squares
𝑎! − 𝑏 ! = (𝑎 + 𝑏)(𝑎 − 𝑏)
𝑥 ! − 25 = (𝑥 + 5)(𝑥 − 5)
Difference of Cubes
𝑎! − 𝑏! =
(𝑎 − 𝑏)(𝑎! + 𝑎𝑏 + 𝑏 ! )
𝑥 ! − 27 =
(𝑥 − 3)(𝑥 ! + 3𝑥 + 9)
Sum of Cubes
𝑎! + 𝑏! =
(𝑎 + 𝑏)(𝑎! − 𝑎𝑏 + 𝑏 ! )
𝑥 ! + 64 =
(𝑥 + 4)(𝑥 ! − 4𝑥 + 16)
𝑥 ! + 6𝑥 + 9 = (𝑥 + 3)!
𝑥 ! + 12𝑥 + 36 = 𝑥 + 6
𝑥 ! − 8𝑥 + 16 = (𝑥 − 4)!
𝑎𝑥 + 𝑏𝑥 + 𝑎𝑦 + 𝑏𝑦 =
𝑥 ! − 10𝑥 + 25 = (𝑥 − 5)!
4𝑥 + 16𝑥𝑧 + 3𝑦 + 12𝑦𝑧 =
𝑥 𝑎+𝑏 +𝑦 𝑎+𝑏 =
4𝑥 1 + 𝑧 + 3𝑦 1 + 𝑧 =
(𝑥 + 𝑦)(𝑎 + 𝑏)
(4𝑥 + 3𝑦)(1 + 𝑧)
Perfect Square Trinomials
!
Grouping
4 OR MORE
Example 1: Factor each polynomial. If the polynomial cannot be factored, write prime.
A) x6 – y6
B) a3x2 – 6a3x + 9a3 – b3x2 + 6b3x - 9b3
C) a6 + b6
D) x5 + 4x4 + 4x3 + x2y3 + 4xy3 + 4y3
E) x3 + x2 – x – 1 F) 3x3 – 9x2 + 4x – 12
G) x3 + 8 I) 16x4 + 54xy3 H) 27y3 + 125 J) 9y3 + 5x3 Example 2: Solve the following equations. A) 18x4 – 21x2 + 3 =0 B) 8x4 + 10x2 – 12 = 0 5E I can find all of the zeros of a polynomial function. Possible roots can be determined by looking at the factors of the constant term and leading coefficient, and
can be written as rational fractions – these are the possible RATIONAL ROOTS. This doesn’t include
irrational or complex roots that may exist:
!"#$%&' !" !"#$%&#% !"#$
!"#$%&' !" !"#$%&' !"#$$%!%#&'
Example 1: List all of the possible rational zeros of each function.
A) f(x) = 4x5 + x4 – 2x3 -­‐ 5x2 + 8x + 16 B) f(x) = x3 – 2x2 + 5x + 12 We use SYNTHETIC DIVISION to work towards identifying all of the roots. You must repeat synthetic
division until you have a polynomial of degree 2 that can you can factor or use the Quadratic Formula with.
Example 2: Find all of the zeros of the following functions.
B) f(x) = 𝑥 ! + 9𝑥 ! + 6𝑥 − 16
A) f(x) = 2x4 – 5x3 + 20x2 – 45x + 18
C) f(x) = 5x4 – 8x3 + 41x2 – 72x – 36. D) f(x) = 5x4 – 29x3 + 55x2 – 28x