Honors Algebra 2 Final Exam Review
38 Questions
101 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the value of $i^1$?

  • -1
  • 1
  • 0
  • i (correct)

What is the value of $i^2$?

-1

What is the value of $i^3$?

-i

What is the value of $i^4$?

<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

Flashcards

What is the value of i^1 ?

i raised to the power of 1 equals i

What is the value of i^2 ?

i raised to the power of 2 equals -1

What is the value of i^3 ?

i raised to the power of 3 equals -i

What is the value of i^4 ?

i raised to the power of 4 equals 1

Signup and view all the flashcards

What does (X^m)^n equal ?

To raise a power to another power, multiply the exponents.

Signup and view all the flashcards

What does X^m X^n equal ?

When multiplying exponents with the same base, add the powers.

Signup and view all the flashcards

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

To divide exponents with the same base, subtract the powers.

Signup and view all the flashcards

What does (XY)^m equal ?

To raise a product to a power, raise each factor to that power.

Signup and view all the flashcards

What does (X/Y)^m equal ?

To raise a quotient to a power, raise both the numerator and denominator to that power.

Signup and view all the flashcards

What does X^(-n) equal ?

X raised to the power of negative n equals 1 divided by X raised to the power of n.

Signup and view all the flashcards

What is the value of 2^2 ?

2 raised to the power of 2 equals 4

Signup and view all the flashcards

What is the value of 2^3 ?

2 raised to the power of 3 equals 8

Signup and view all the flashcards

What is the value of 2^4 ?

2 raised to the power of 4 equals 16

Signup and view all the flashcards

What is the value of 2^5 ?

2 raised to the power of 5 equals 32

Signup and view all the flashcards

What is the value of 2^6 ?

2 raised to the power of 6 equals 64

Signup and view all the flashcards

What is the value of 2^7 ?

2 raised to the power of 7 equals 128

Signup and view all the flashcards

What is the value of 2^8 ?

2 raised to the power of 8 equals 256

Signup and view all the flashcards

What is the value of 2^9 ?

2 raised to the power of 9 equals 512

Signup and view all the flashcards

What is the value of 2^10 ?

2 raised to the power of 10 equals 1024

Signup and view all the flashcards

What is the value of 3^3 ?

3 raised to the power of 3 equals 27

Signup and view all the flashcards

What is the value of 3^4 ?

3 raised to the power of 4 equals 81

Signup and view all the flashcards

What is the value of 3^5 ?

3 raised to the power of 5 equals 243

Signup and view all the flashcards

What is the value of 6^3 ?

6 raised to the power of 3 equals 216

Signup and view all the flashcards

What is the value of 5^4 ?

5 raised to the power of 4 equals 625

Signup and view all the flashcards

What is the value of 7^3 ?

7 raised to the power of 3 equals 343

Signup and view all the flashcards

What is the value of 9^3 ?

9 raised to the power of 3 equals 729

Signup and view all the flashcards

What is the value of 12^3 ?

12 raised to the power of 3 equals 1728

Signup and view all the flashcards

What is the Remainder Theorem?

When a polynomial f(x) is divided by x-k, the remainder is equal to f(k).

Signup and view all the flashcards

What is the Factor Theorem?

If the value of a polynomial f(x) is equal to zero when x=k, then (x-k) is a factor of the polynomial.

Signup and view all the flashcards

What is the Rational Zeros Theorem?

If (ax-b) is a factor of a polynomial f(x), then f(a/b) equals zero.

Signup and view all the flashcards

What is the Fundamental Theorem of Algebra?

A polynomial of degree n has exactly n complex zeros, counting multiplicities.

Signup and view all the flashcards

What is the Linear Factorization Theorem?

Every polynomial can be factored into n linear binomials, where each binomial is of the form (x-z), where z is a zero of the polynomial.

Signup and view all the flashcards

What is the Complex Conjugate Theorem?

If a polynomial with real coefficients has non-real zeros, then those zeros come in conjugate pairs.

Signup and view all the flashcards

What is the Upper Bound Theorem?

If K is greater than 0 and all the coefficients in the quotient of synthetic division are positive, then K is an upper bound for the real zeros of the polynomial.

Signup and view all the flashcards

What is the Lower Bound Theorem?

If K is less than 0 and the coefficients in the quotient of synthetic division alternate between positive and negative (or zero), then K is a lower bound for the real zeros of the polynomial.

Signup and view all the flashcards

What is the explicit formula for a geometric sequence?

The explicit formula for a geometric sequence is Tn=T1 x r^(n-1) where Tn is the nth term, T1 is the first term, r is the common ratio, and n is the term number.

Signup and view all the flashcards

What is the explicit formula for an arithmetic sequence?

The explicit formula for an arithmetic sequence is An = A1 + D (n-1) where An is the nth term, A1 is the first term, D is the common difference, and n is the term number.

Signup and view all the flashcards

If _____________ and it is geometric then it converges

If the absolute value of the common ratio (r) of a geometric sequence is less than 1, and it is geometric then it converges.

Signup and view all the flashcards

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 ).

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

Description

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.

More Like This

Quadratic Formula and Imaginary Numbers Quiz
10 questions

Quadratic Formula and Imaginary Numbers Quiz

IdyllicPrairie avatar
IdyllicPrairie
Real and Imaginary Numbers Quiz
18 questions

Real and Imaginary Numbers Quiz

SolicitousPelican7010 avatar
SolicitousPelican7010
Real and Imaginary Numbers Quiz
14 questions

Real and Imaginary Numbers Quiz

PreEminentSaxophone avatar
PreEminentSaxophone
Imaginary Numbers in Applied Mathematics
18 questions

Imaginary Numbers in Applied Mathematics

UnmatchedSymbolism7563 avatar
UnmatchedSymbolism7563
Use Quizgecko on...
Browser
Open
Browser
Browser