Questions and Answers
What is the value of $i^1$?
What is the value of $i^2$?
-1
What is the value of $i^3$?
-i
What is the value of $i^4$?
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What does $(X^m)^n$ equal?
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What does $X^m X^n$ equal?
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What does $(X^m)/(X^n)$ equal?
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What does $(XY)^m$ equal?
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What does $(X/Y)^m$ equal?
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What does $X^{-n}$ equal?
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What is the value of $2^2$?
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What is the value of $2^3$?
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What is the value of $2^4$?
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What is the value of $2^5$?
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What is the value of $2^6$?
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What is the value of $2^7$?
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What is the value of $2^8$?
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What is the value of $2^9$?
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What is the value of $2^{10}$?
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What is the value of $3^3$?
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What is the value of $3^4$?
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What is the value of $3^5$?
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What is the value of $6^3$?
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What is the value of $5^4$?
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What is the value of $7^3$?
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What is the value of $9^3$?
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What is the value of $12^3$?
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What is the Remainder Theorem?
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What is the Factor Theorem?
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What is the Rational Zeros Theorem?
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What is the Fundamental Theorem of Algebra?
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What is the Linear Factorization Theorem?
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What is the Complex Conjugate Theorem?
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What is the Upper Bound Theorem?
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What is the Lower Bound Theorem?
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What is the explicit formula for a geometric sequence?
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What is the explicit formula for an arithmetic sequence?
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If _____________ and it is geometric then it converges
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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.