Algebra Archive Study Guide Trimmed Algebra 2 Review
Semester 2 Final Study Guide
NAME:
DATE:
1. Radicals & Rational Exponents
Simplifying Radicals
Simplify completely:
1. \(\sqrt{72x^5y^8}\)
2. \(\sqrt[3]{54a^4b^9}\)
Operations with Radicals
Simplify and combine:
3. \(3\sqrt{20} + \sqrt{45} - 2\sqrt{80}\)
4. \((2 + \sqrt{3})(5 - 2\sqrt{3})\)
Converting Radicals & Exponents
Write in radical form:
5. \((16x)^{3/4}\)
Write in rational exponent form:
6. \(\sqrt[5]{x^2y^7}\)
Radical Equations
Solve for \(x\):
7. \(\sqrt{2x + 5} = 9\)
8. \(\sqrt[3]{x - 4} + 2 = 5\)
Page 1 of 5
2. Complex Numbers
Simplify and combine:
9. \((7 - 3i) - (4 + 10i)\)
Multiply and simplify:
10. \((2 + 5i)(3 - i)\)
3. Factoring Polynomials (Part I)
GCF & Factoring with GCF
11. \(12x^4 - 18x^2\)
12. \(5x^3y + 10x^2y^2 - 15xy^3\)
Trinomials (\(a=1\))
13. \(x^2 - 7x - 18\)
14. \(x^2 + 11x + 24\)
Page 2 of 5
3. Factoring Polynomials (Part II)
Trinomials (\(a > 1\))
15. \(3x^2 + 13x - 10\)
16. \(2x^2 - 7x + 3\)
Difference of Squares
17. \(16x^2 - 81\)
18. \(4x^2 - 25y^2\)
Sum and Difference of Cubes
19. \(x^3 + 27\)
20. \(8x^3 - 125\)
Page 3 of 5
4. Parent Functions & Quadratics
Identifying Parent Functions
Identify the parent function (Linear, Quadratic, Cubic, Absolute Value, Quartic, or Quintic):
21. \(f(x) = 3x - 5\)
22. \(g(x) = x^2 + 4x + 4\)
23. \(h(x) = |x - 2| + 5\)
24. \(j(x) = -x^3 + 8\)
25. \(k(x) = x^4 - 2x^2\)
26. \(l(x) = x^5 + 1\)
Solving Quadratics
Solve by Factoring:
27. \(x^2 - 5x + 6 = 0\)
28. \(x^2 + 2x - 15 = 0\)
Solve using Quadratic Formula:
29. \(x^2 + 4x - 1 = 0\)
30. \(2x^2 - 3x - 5 = 0\)
Page 4 of 5
Solving from a Graph
31. Find solutions for \(f(x)=0\):
\(x \approx\) __________
32. Find solutions for \(g(x)=0\):
\(x \approx\) __________
5. Rational Functions
Asymptotes & Operations
Asymptotes:
33. \(f(x) = \frac{3}{x - 4} + 2\)
34. \(g(x) = \frac{2x + 1}{x + 5}\)
Simplify:
35. \(\frac{x^2 - 9}{x + 3} \cdot \frac{5}{x - 3}\)
36. \(\frac{x^2 + 2x - 8}{x^2 + 4x} \div \frac{x - 2}{x}\)
Algebra Archive Semester 2 Final Study Guide
Page 5 of 5
Algebra Archive Cheat Sheet Algebra II Cheat Sheet
Semester 2 Quick Reference Guide
v2.0.26
Factoring Toolkit
1. GCF: Always check first! \(ab + ac = a(b+c)\)
2. Difference of Squares: \(a^2 - b^2 = (a-b)(a+b)\)
3. Sum/Diff of Cubes:
\(a^3 + b^3 = (a+b)(a^2-ab+b^2)\)
Mnemonic: SOAP (Same, Opposite, Always Positive)
4. Trinomials \(a=1\): \(x^2+bx+c \rightarrow\) Find factors of \(c\) that add to \(b\).
5. Trinomials \(a>1\): Use AC method or "Slide and Divide".
Radicals & Complex
Rational Exponents
\(x^{m/n} = \sqrt[n]{x^m}\)
Complex Units
\(i = \sqrt{-1}\)
\(i^2 = -1\)
\(i^3 = -i\)
\(i^4 = 1\)
Quadratic Formula
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Parent Function Field Guide
Linear
\(f(x)=x\)
Quadratic
\(f(x)=x^2\)
Abs Value
\(f(x)=|x|\)
Cubic
\(f(x)=x^3\)
Quartic
\(f(x)=x^4\)
Quintic
\(f(x)=x^5\)
Rational Express Logic
Asymptotes:
VA: Where denominator = 0
HA: Compare degrees of top (\(n\)) and bottom (\(m\))
If \(n < m\), HA is \(y=0\)
If \(n = m\), HA is \(y = \text{ratio of leading coeffs}\)
Operations:
Mult: Factor everything, then cancel common factors.
Div: Keep, Change, Flip. Factor, then cancel.
Watch out for excluded values!
Algebra Archive Slides Algebra II
Final Review
Semester 2 Archive
Radicals
Factoring
Functions
Radical Refresh
Simplifying Strategy
• Pull out perfect powers (\(n\)-th roots)
• Variables: Divide exponent by index
• Remainder stays inside!
\[\sqrt[n]{x^m} = x^{m/n}\]
Challenge Problem
\(\sqrt[3]{128x^7y^3}\)
Factoring Flow
Level 1: GCF
Always the first step. Look for the biggest shared factor.
Level 2: Trinomials
Find numbers that multiply to \(ac\) and add to \(b\).
Level 3: Cubes
SOAP:
\(8x^3 + 125\)
Factor it!
The Family Reunion
ABSOLUTE VALUE
\(y = |x|\)
CUBIC
\(y = x^3\)
QUARTIC
\(y = x^4\)
QUINTIC
\(y = x^5\)
Rational Rundown
Multiplication
Factor numerators and denominators, then cancel anything shared!
Division
\[\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}\]
PRACTICE THOUGHTS
\(\frac{x^2 - 16}{x + 4} \cdot \frac{1}{x - 4}\)
What happens to the terms?
You Got This
Don't forget to check your signs!
1
Factor
2
Check for
3
Simplify
Algebra Archive Cheat Sheet Practice Sheet Algebra II Reference
Semester 2 Quick Logic Archive
REF.S2
Factoring Toolkit
GCF: \(ab + ac = a(b+c)\)
Diff Squares: \(a^2 - b^2 = (a-b)(a+b)\)
Cubes: SOAP (Same, Opp, Always Pos)
\(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
Trinomials: find factors of \(ac\) that sum to \(b\).
Radicals & Complex
\(x^{m/n} = \sqrt[n]{x^m}\)
\(i = \sqrt{-1}\)
\(i^2 = -1\)
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Parent Functions
Linear
Quad
Cubic
Abs Val
Quartic
Quintic
Rationals
Asymptotes:
VA: Denominator = 0
HA: Top degree vs Bot degree
Operations:
Mult: Factor + Cancel
Div: Keep-Change-Flip then Mult
Archive Practice (2x Each)
Radicals
1. Simplify \(\sqrt{48x^3}\)
2. Simplify \(\sqrt[3]{16a^4}\)
3. Solve \(\sqrt{x+1} = 5\)
4. Solve \(\sqrt{3x} = 6\)
Complex
5. \((2+i) + (3-2i)\)
6. \((1+i)(1-i)\)
Factoring
7. \(x^2 + 5x + 6\)
8. \(2x^2 + 7x + 3\)
Cubes/Special
9. \(x^3 - 8\)
10. \(x^3 + 1\)
11. \(x^2 - 25\)
12. \(4x^2 - 9\)
Quadratics
13. Solve \(x^2 = 16\)
14. Quad Formula \(x^2 + 2x - 4 = 0\)
Rationals
15. VA of \(y = \frac{1}{x-2}\)
16. VA of \(y = \frac{1}{x+3}\)
END OF QUICK REFERENCE ARCHIVE
Algebra Archive Slides Final Practice Deck Algebra II
Final Review
Semester 2 Archive
Radical Refresh
Simplifying Strategy
\[\sqrt[n]{x^m} = x^{m/n}\]
• Pull out perfect powers
• Exponents: Divide by index
Practice 1
\(\sqrt{72x^5}\)
Practice 2
\(\sqrt[3]{54a^4}\)
Complex Units
Remember: \(i = \sqrt{-1}\) and \(i^2 = -1\).
Example: \((2+i)(3-i)\)
FOIL: \(6 - 2i + 3i - i^2\)
Result: \(7 + i\)
P1: Addition
\((7 - 3i) - (4 + 2i)\)
P2: Multi
\((5 + 2i)(5 - 2i)\)
Factoring Flow
Standard Form: \(ax^2 + bx + c\)
Problem 1: \(a=1\)
\(x^2 - 7x - 18\)
Problem 2: Cubes
\(x^3 + 27\)
SOAP Reminder
Sum/Diff of Cubes uses:
Quadratic Solvers
Quadratic Formula
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
P1: Solve \(x^2 + 4x - 1 = 0\)
Solving from Graph
P2: Roots are where the graph hits the x-axis.
The Family Reunion
QUADRATIC
\(y = x^2\)
CUBIC
\(y = x^3\)
QUARTIC
\(y = x^4\)
Rational Rundown
Practice 1: Mult
\(\frac{x^2-9}{x+3} \cdot \frac{5}{x-3}\)
Practice 2: Div
\(\frac{x+2}{x} \div \frac{x^2-4}{x}\)
Quick Tips
1. Factor everything first!
2. For Division: Flip second term
3. Cancel shared factors
4. Note excluded values
Algebra Archive Study Guide Trimmed Final Algebra 2 Review
Semester 2 Final Study Guide
NAME:
DATE:
1. Radicals & Rational Exponents
Simplifying Radicals
Simplify completely:
1. \(\sqrt{72x^5y^8}\)
2. \(\sqrt[3]{54a^4b^9}\)
Operations with Radicals
Simplify and combine:
3. \(3\sqrt{20} + \sqrt{45} - 2\sqrt{80}\)
4. \((2 + \sqrt{3})(5 - 2\sqrt{3})\)
Converting Radicals & Exponents
Write in radical form:
5. \((16x)^{3/4}\)
Write in rational exponent form:
6. \(\sqrt[5]{x^2y^7}\)
Radical Equations
Solve for \(x\):
7. \(\sqrt{2x + 5} = 9\)
8. \(\sqrt[3]{x - 4} + 2 = 5\)
Page 1 of 5
2. Complex Numbers
Addition / Subtraction:
9. \((7 - 3i) - (4 + 10i)\)
Multiplication:
10. \((2 + 5i)(3 - i)\)
3. Factoring Polynomials
GCF & Factoring with GCF
11. \(12x^4 - 18x^2\)
12. \(5x^3y + 10x^2y^2 - 15xy^3\)
Trinomials (\(a=1\))
13. \(x^2 - 7x - 18\)
14. \(x^2 + 11x + 24\)
Page 2 of 5
3. Factoring Polynomials (Continued)
Trinomials (\(a > 1\))
15. \(3x^2 + 13x - 10\)
16. \(2x^2 - 7x + 3\)
Special Cases
Difference of Squares:
17. \(16x^2 - 81\)
Difference of Squares:
18. \(4x^2 - 25y^2\)
Sum and Difference of Cubes
19. \(x^3 + 27\)
20. \(8x^3 - 125\)
Page 3 of 5
4. Parent Functions & Quadratics
Identifying Parent Functions
21. \(f(x) = 3x - 5\)
22. \(g(x) = x^2 + 4x + 4\)
23. \(h(x) = |x - 2| + 5\)
24. \(j(x) = -x^3 + 8\)
25. \(k(x) = x^4 - 2x^2\)
26. \(l(x) = x^5 + 1\)
Solving Quadratics (Large Work Area)
Solve by Factoring:
27. \(x^2 - 5x + 6 = 0\)
Solve using Quadratic Formula:
28. \(x^2 + 4x - 1 = 0\)
Page 4 of 5
Solving from a Graph
29. Identify solutions (\(f(x)=0\)):
\(x \approx\) __________
30. Identify solutions (\(g(x)=0\)):
\(x \approx\) __________
Algebra Archive Cheat Sheet Practice Repaired Algebra II Reference
Semester 2 Quick Logic Archive
S2-REF
Factoring Toolkit
GCF: \(ab + ac = a(b+c)\)
Diff Squares: \(a^2 - b^2 = (a-b)(a+b)\)
Cubes: SOAP (Same, Opp, Always Pos)
\(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
Trinomials: Find \(x \cdot y = ac\) and \(x + y = b\).
Radicals & Complex
\(x^{m/n} = \sqrt[n]{x^m}\)
\(i = \sqrt{-1}\)
\(i^2 = -1\)
\(x = \displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Parent Function Archive
Linear
Quad
Abs Val
Cubic
Quartic
Quintic
Rational Rules
Asymptotes:
VA: Set denominator = 0.
HA: Top deg < Bot deg \(\rightarrow y = 0\)
HA: Top deg = Bot deg \(\rightarrow y = \text{Ratio coeffs}\)
Logic:
Mult: Factor, then cancel common terms.
Div: Multiply by reciprocal (Flip the 2nd).
Archive Practice (2x Each)
1. Radicals
A. Simplify \(\sqrt{48x^3}\)
B. Solve \(\sqrt{x+1} = 5\)
2. Complex
A. \((2+i) - (3-2i)\)
B. \((1+i)(1-i)\)
3. Factoring
A. Trinomial \(x^2 + 5x + 6\)
B. Cubes \(x^3 - 8\)
4. Functions
A. Solve \(x^2 + 2x - 4 = 0\)
B. VA of \(y = \frac{1}{x-2}\)
Algebra Archive Practice Vault — Complete 2 per Section
Algebra Archive Slides Definitive Final Algebra II
Final Review
Semester 2 Archive
Radical Foundations
Rules of Engagement
\(x^{m/n} = \sqrt[n]{x^m}\)
• Simplify: Pull out perfect \(n\)-th powers
• Equations: Isolate the radical, then power up!
Practice 1: Simplify
\(\sqrt{72x^5y^8}\)
Practice 2: Convert
\((16a)^{3/4}\)
The Complex Archive
Key Constant: \(i = \sqrt{-1}\)
Operations:
• Add: Combine like terms
• Mult: FOIL, then replace \(i^2\)
\((7 - 3i) - (4 + 10i)\)
\((2 + 5i)(3 - i)\)
Factoring Masterclass
Level 1: GCF & Trinomials
\(ax^2 + bx + c\)
\(12x^4 - 18x^2\)
\(x^2 - 7x - 18\)
Level 2: Cubes
\(x^3 + 27\)
Remember SOAP: Same, Opposite, Always Positive!
Quadratic Solvers
Quadratic Formula
\(x = \displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Solve: \(x^2 + 4x - 1 = 0\)
Graphic Solutions
Solutions are the \(x\)-intercepts!
Function Family Archive
ABSOLUTE VALUE
\(y = |x|\)
CUBIC
\(y = x^3\)
QUARTIC
\(y = x^4\)
Rational Rundown
The Strategy
1. Factor every numerator
2. Factor every denominator
3. Flip divisors (Division only)
4. Cancel shared factors
Practice Challenge:
\(\displaystyle \frac{x^2 - 9}{x + 3} \cdot \frac{5}{x - 3}\)
Solve mentally: What factors remain?
Algebra Archive Cheat Sheet Practice Repaired Final Algebra II Reference
Semester 2 Quick Logic Archive
S2-ARCHIVE
Factoring Toolkit
GCF: \(ab + ac = a(b+c)\)
Diff Squares: \(a^2 - b^2 = (a-b)(a+b)\)
Cubes: SOAP (Same, Opp, Always Pos)
\(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
Trinomials: Find \(x \cdot y = ac\) and \(x + y = b\).
Radicals & Complex
\(x^{m/n} = \sqrt[n]{x^m}\)
\(i = \sqrt{-1}\)
\(i^2 = -1\)
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Parent Function Archive
Linear
Quad
Abs Val
Cubic
Quartic
Quintic
Rational Essentials
Asymptotes:
VA: Denominator = 0
HA (deg \(n/m\)): \(n
Pro-Tips:
Factor every term completely BEFORE canceling.
Multiply by reciprocal for division operations.
Archive Practice (2x Each)
1. Radicals
A. Simplify \(\sqrt{48x^3}\)
B. Solve \(\sqrt{x+1} = 5\)
2. Complex
A. \((2+i) - (3-2i)\)
B. \((1+i)(1-i)\)
3. Factoring
A. Trinomial \(x^2 + 5x + 6\)
B. Cubes \(x^3 - 8\)
4. Functions
A. Solve \(x^2 + 2x - 4 = 0\)
B. VA of \(y = \frac{1}{x-2}\)
End of Algebra II Final Review Archive
Algebra Archive Slides Definitive Final Repaired Algebra II
Final Review
Semester 2 Archive
Radical Essentials
Conversion Key
\(x^{m/n} = \sqrt[n]{x^m}\)
1. Write in Radical Form:
\((16x)^{3/4}\)
2. Write in Expo Form:
\(\sqrt[5]{x^2y^7}\)
3. Radical Equations
\(\sqrt{2x + 5} = 9\)
4. Complex Radicals
\(\sqrt[3]{x - 4} + 2 = 5\)
The Complex Archive
Identity: \(i = \sqrt{-1}\)
Strategies:
• Treat \(i\) as a variable (\(x\))
• Simplify \(i^2\) immediately to \(-1\)
• Result format: \(a + bi\)
Practice 1: Add/Sub
\((7 - 3i) - (4 + 10i)\)
Practice 2: Multiply
\((2 + 5i)(3 - i)\)
Factoring Masterclass
I. Basic Factors
GCF:
\(12x^4 - 18x^2\)
Trinomial (\(a=1\)):
\(x^2 - 7x - 18\)
II. Advanced Factors
Trinomial (\(a>1\)):
\(3x^2 + 13x - 10\)
Difference Squares:
\(16x^2 - 81\)
Advanced Solvers
Cubes (SOAP)
\(x^3 + 27\)
\(8x^3 - 125\)
Same, Opposite, Always Positive!
Quad Formula
\(x = \displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
P1: \(x^2 + 4x - 1 = 0\)
P2: \(2x^2 - 3x - 5 = 0\)
Function Family Archive
LINEAR
\(y = x\)
QUADRATIC
\(y = x^2\)
ABS VALUE
\(y = |x|\)
CUBIC
\(y = x^3\)
QUARTIC
\(y = x^4\)
QUINTIC
\(y = x^5\)
Rational Rundown
Asymptote Check
• VA: Set Denominator = 0
• HA: Compare Degrees \(n\) vs \(m\)
If \(n = m \rightarrow y = \text{ratio}\)
Practice 1: Mult
\(\displaystyle \frac{x^2 - 9}{x + 3} \cdot \frac{5}{x - 3}\)
Practice 2: Div
\(\displaystyle \frac{x^2 + 2x - 8}{x^2 + 4x} \div \frac{x - 2}{x}\)
