Algebra II
Chapter 2 Study Guide
Without studying this guide is useless
2.1 Use Properties of Exponents
Product of Powers: 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
Power of a Power: (𝑎𝑚 )𝑛 = 𝑎𝑚∗𝑛
Power of a Product: (𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚
1
Negative Exponent: 𝑎 −𝑚 = 𝑎𝑚 , 𝑎 ≠ 0
Zero Power: 𝑎0 = 1, 𝑎 ≠ 0
Quotient of a Power:
𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛 , 𝑎 ≠ 0
𝑎
𝑎𝑚
Power of a Quotient: (𝑏)𝑚 = 𝑏𝑚 , 𝑎 ≠ 0
Scientific Notation: 𝑎 × 10𝑚 , 1 < 𝑎 < 10
Simplified Form: No negative exponents, reduced exponents, simplified variables
2.2 Evaluate and Graph Polynomial Functions
Degree: The highest exponent in a polynomial function
Leading Coefficient: The number attached to the variable with the highest
Exponent in a polynomial function.
Constant Term: Term in a polynomial function that has no variable
Synthetic Substitution: A method of finding values of a polynomial function
by using the coefficients.
End Behavior: Given the Degree (Odd/Even) and sign of the leading
Coefficient, you can determine the general shape of the function.
2.3 Add, Subtract, and Multiply Polynomials
Like terms: Terms with the same variable and same power.
Add / Subtract with the vertical and Horizontal method
Multiply with the Vertical and Horizontal Method
Special Types (Key Concept Box pg. 105)
2.4 Factor and Solve Polynomial Equations
Factored Completely: A function written as a product of unfactorable
polynomials with integer coefficients.
Sum of two cubes: 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )
Difference of Two Cubes: 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 )
Factor by Grouping:
𝑎𝑥 + 𝑎𝑦 + 𝑏𝑥 + 𝑏𝑦 = 𝑎(𝑥 + 𝑦) + 𝑏(𝑥 + 𝑦) = (𝑎 + 𝑏)(𝑥 + 𝑦)
Factor When in Quadratic form: (𝑎𝑚 )2 + 𝑏 𝑚 + 𝑐
2.5 Applying the Remainder and Factor Theorem
Polynomial Long Division: Dividing polynomials through long division
Synthetic Division: Division using the coefficients where the quotient is the
coefficients of the polynomial.
Remainder Theorem: if 𝑓(𝑥) ix divided by 𝑥 − 𝑘, then the remainder is 𝑟 = 𝑓(𝑘)
Factor Theorem: a polynomial 𝑓(𝑥)has a factor 𝑥 − 𝑘, if and only if 𝑓(𝑘) = 0
Zeros: the value k when 𝑥 − 𝑘 is a factor of 𝑓(𝑥)
2.6 Find Rational Zeros
Rational Zero Theorem If 𝑓(𝑥) has integer coefficients then every rational
zero of has the form of
𝑝
𝑞
=
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑒𝑐𝑖𝑒𝑛𝑡
Possible Rational Zeros: all the positive and negative values of p and q
Using Rational Theorem to find factors of 𝑓(𝑥) (synthetic division)
2.7 Apply the fundamental Theorem of Algebra
Repeated Solution: solutions that appear more than once in a factored form
of 𝑓(𝑥); squared appears twice, cubed appears three times.
Fundamental Theorem of Algebra: if 𝑓(𝑥) is a polynomial of degree n, n > 0
then 𝑓(𝑥) = 0 has at least one solution in the set of complex numbers
Corollary: If 𝑓(𝑥) is a polynomial of degree n, n > 0 then 𝑓(𝑥) = 0 has exactly
n solutions where repeated solutions are counted as such.
Complex Conjugates Theorem: if 𝑎 + 𝑏𝑖 is an imaginary zero then 𝑎 − 𝑏𝑖 is
also an imaginary solution
Irrational Conjugates Theorem: If 𝑎 + √𝑏 is a zero of 𝑓(𝑥) then 𝑎 − √𝑏 is also
a zero of the function
Descarteś Rule of Signs: A way of determining the possible number of
positive and negative zeros of a function
Positive: Count the number of sign changes for 𝑓(𝑥)
Negative: Count the number of sign changes for 𝑓(−𝑥)
2.8 Analyze Graphs of Polynomial Functions
Turning Points: Local Minimum Local Maximum
points
The graph of every polynomial function of degree n has at most n-1 turning
If a polynomial function has n distinct real zeros, then the graph has exactly
n-1 turning points