Honors Algebra 2 Final Exam Review

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Questions and Answers

What is the value of $i^1$?

-1

i (correct)

What is the value of $i^2$?

-1

What is the value of $i^3$?

-i

What is the value of $i^4$?

1 Signup and view all the answers

What does $(X^m)^n$ equal?

X^(mn) Signup and view all the answers

What does $X^m X^n$ equal?

X^(m+n) Signup and view all the answers

What does $(X^m)/(X^n)$ equal?

X^(m-n) Signup and view all the answers

What does $(XY)^m$ equal?

X^m Y^m Signup and view all the answers

What does $(X/Y)^m$ equal?

(X^m)/(Y^m) Signup and view all the answers

What does $X^{-n}$ equal?

(1/X^n) Signup and view all the answers

What is the value of $2^2$?

4 Signup and view all the answers

What is the value of $2^3$?

8 Signup and view all the answers

What is the value of $2^4$?

16 Signup and view all the answers

What is the value of $2^5$?

32 Signup and view all the answers

What is the value of $2^6$?

64 Signup and view all the answers

What is the value of $2^7$?

128 Signup and view all the answers

What is the value of $2^8$?

256 Signup and view all the answers

What is the value of $2^9$?

512 Signup and view all the answers

What is the value of $2^{10}$?

1024 Signup and view all the answers

What is the value of $3^3$?

27 Signup and view all the answers

What is the value of $3^4$?

81 Signup and view all the answers

What is the value of $3^5$?

243 Signup and view all the answers

What is the value of $6^3$?

216 Signup and view all the answers

What is the value of $5^4$?

625 Signup and view all the answers

What is the value of $7^3$?

343 Signup and view all the answers

What is the value of $9^3$?

729 Signup and view all the answers

What is the value of $12^3$?

1728 Signup and view all the answers

What is the Remainder Theorem?

If f(x) is divided by x-k then the remainder is f(k) Signup and view all the answers

What is the Factor Theorem?

If f(k)=0 then (x-k) is a factor Signup and view all the answers

What is the Rational Zeros Theorem?

If (ax-b) is a factor of f(x) then f(a/b)=0 Signup and view all the answers

What is the Fundamental Theorem of Algebra?

An nth degree polynomial has n complex zeros Signup and view all the answers

What is the Linear Factorization Theorem?

A polynomial can factor into n linear binomials where each binomial has the form (x-z) Signup and view all the answers

What is the Complex Conjugate Theorem?

A polynomial with Real Coefficients will have non-real zeros come in conjugate pairs Signup and view all the answers

What is the Upper Bound Theorem?

If K > 0, and you get all positive coefficients in the quotient of your synthetic division then k is an upper bound for the real zeros Signup and view all the answers

What is the Lower Bound Theorem?

If K < 0 and you get alternating positive and negative coefficients in the quotient of your synthetic division then k is a lower bound for the real zeros Signup and view all the answers

What is the explicit formula for a geometric sequence?

Tn=T1 x r^(n-1) Signup and view all the answers

What is the explicit formula for an arithmetic sequence?

An = A1 + D (n-1) Signup and view all the answers

If _____________ and it is geometric then it converges

-1 Signup and view all the answers

Study Notes

Powers of i

( i^1 = i )
( i^2 = -1 )
( i^3 = -i )
( i^4 = 1 )

Exponent Rules

( (X^m)^n = X^{mn} )
( X^m \cdot X^n = X^{m+n} )
( \frac{X^m}{X^n} = X^{m-n} )
( (XY)^m = X^m \cdot Y^m )
( \left(\frac{X}{Y}\right)^m = \frac{X^m}{Y^m} )
( X^{-n} = \frac{1}{X^n} )

Exponential Values

( 2^2 = 4 )
( 2^3 = 8 )
( 2^4 = 16 )
( 2^5 = 32 )
( 2^6 = 64 )
( 2^7 = 128 )
( 2^8 = 256 )
( 2^9 = 512 )
( 2^{10} = 1024 )
( 3^3 = 27 )
( 3^4 = 81 )
( 3^5 = 243 )
( 6^3 = 216 )
( 5^4 = 625 )
( 7^3 = 343 )
( 9^3 = 729 )
( 12^3 = 1728 )

Theorems in Algebra

Remainder Theorem: For polynomial ( f(x) ) divided by ( x-k ), remainder equals ( f(k) ).
Factor Theorem: If ( f(k) = 0 ), then ( (x-k) ) is a factor of ( f(x) ).
Rational Zeros Theorem: For polynomial ( ax^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 ), potential rational zeros are given by the ratio of factors of ( a_0 ) to factors of ( a_n ).
Fundamental Theorem of Algebra: An ( n )-degree polynomial has ( n ) complex zeros.
Linear Factorization Theorem: An ( n )-degree polynomial can be expressed as ( n ) linear binomials of the form ( (x-z) ).
Complex Conjugate Theorem: Non-real zeros of polynomials with real coefficients appear in conjugate pairs.
Upper Bound Theorem: If ( K > 0 ) and the synthetic division yields all positive coefficients, then ( K ) is an upper bound for the real zeros.
Lower Bound Theorem: If ( K < 0 ) and the synthetic division yields alternating coefficients, then ( K ) is a lower bound for real zeros.

Sequences

Explicit Formula for Geometric Sequence: ( T_n = T_1 \cdot r^{(n-1)} )
Explicit Formula for Arithmetic Sequence: ( A_n = A_1 + D(n-1) )
A geometric sequence converges if the common ratio ( r < -1 ).

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Prepare for your Honors Algebra 2 final exam with these flashcards. Each card covers essential definitions and properties, including imaginary numbers and exponent rules. Use this resource to reinforce your understanding and boost your confidence before the exam.