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Honors Algebra 2 Final Exam Review

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What is the value of $i^4$?

<p>1</p> Signup and view all the answers

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

<p>X^(mn)</p> Signup and view all the answers

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

<p>X^(m+n)</p> Signup and view all the answers

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

<p>X^(m-n)</p> Signup and view all the answers

What does $(XY)^m$ equal?

<p>X^m Y^m</p> Signup and view all the answers

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

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

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

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

What is the value of $2^2$?

<p>4</p> Signup and view all the answers

What is the value of $2^3$?

<p>8</p> Signup and view all the answers

What is the value of $2^4$?

<p>16</p> Signup and view all the answers

What is the value of $2^5$?

<p>32</p> Signup and view all the answers

What is the value of $2^6$?

<p>64</p> Signup and view all the answers

What is the value of $2^7$?

<p>128</p> Signup and view all the answers

What is the value of $2^8$?

<p>256</p> Signup and view all the answers

What is the value of $2^9$?

<p>512</p> Signup and view all the answers

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

<p>1024</p> Signup and view all the answers

What is the value of $3^3$?

<p>27</p> Signup and view all the answers

What is the value of $3^4$?

<p>81</p> Signup and view all the answers

What is the value of $3^5$?

<p>243</p> Signup and view all the answers

What is the value of $6^3$?

<p>216</p> Signup and view all the answers

What is the value of $5^4$?

<p>625</p> Signup and view all the answers

What is the value of $7^3$?

<p>343</p> Signup and view all the answers

What is the value of $9^3$?

<p>729</p> Signup and view all the answers

What is the value of $12^3$?

<p>1728</p> Signup and view all the answers

What is the Remainder Theorem?

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

What is the Factor Theorem?

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

What is the Rational Zeros Theorem?

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

What is the Fundamental Theorem of Algebra?

<p>An nth degree polynomial has n complex zeros</p> Signup and view all the answers

What is the Linear Factorization Theorem?

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

What is the Complex Conjugate Theorem?

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

What is the Upper Bound Theorem?

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

What is the Lower Bound Theorem?

<p>If K &lt; 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</p> Signup and view all the answers

What is the explicit formula for a geometric sequence?

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

What is the explicit formula for an arithmetic sequence?

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

If _____________ and it is geometric then it converges

<p>-1</p> 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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