Algebra 2 Semester 2 Exam Flashcards

Questions and Answers

What are the complex roots of the equation $2x^3 + 4x^4 + 11x - 17 = 0$ given that it has 1 real root at $x=1$?

-3±5i/2

What kind of solutions does $x^5 + 2x^4 + 2x^3 = 0$ have?

3 real, all 0 (correct)

2 imaginary (correct)

x=-1±i (correct)

All of the above

What is the simplified form of $6(4x + 8y) + 2(3x - 6y)$?

30x + 36y

What is the simplified form of $10 ext{√}162 - ext{√}405 - ext{√}200$?

80√2 - 9√5 Signup and view all the answers

What are the solutions for the equation $|x - 2| = 27$?

x=-25 Signup and view all the answers

What is the solution for the system of equations $2x + 5y = 12$ and $x - 3y = -2$?

x = 26/11, y = 16/11 Signup and view all the answers

What is the factorization of $4xy - 3y - 40x + 30$?

(4x - 3)(y - 10) Signup and view all the answers

What is the factorization of $2x^2 + 3x - 9$?

(2x - 3)(x + 3) Signup and view all the answers

What is the result of adding $(5+i)-(6-12i)$?

-1 + 13i Signup and view all the answers

What is the solution for the equation $√{2x - 2} + 12 = 14$?

x = 3 Signup and view all the answers

What is the simplified form of $√{36x^{12}y^8}$?

±6x^6y^4 Signup and view all the answers

What is the simplified form of $(4x^8y^8/2x^3y^{10})^2$?

4x^{10}/y^4 Signup and view all the answers

What is the y-intercept of the function $y = 2 imes 4^x$?

2 Signup and view all the answers

What is the simplified form of $8e^8/4e^{13}$?

2/e^5 Signup and view all the answers

What is the value of $log_2(8)$?

3 Signup and view all the answers

Express $log_r(12) + log_r(45)$ as a single logarithm.

log_r(540) Signup and view all the answers

What is the solution for $1/16 = 64^{3x+5}$?

x = -17/9 Signup and view all the answers

Find x to the nearest 0.01 given $1.95^x = 26$.

x = 4.9 Signup and view all the answers

What is the solution to the equation $e^{0.05t} = 10$?

46.05 Signup and view all the answers

If the number of bacteria, b, after t minutes is given by the equation $b = 10e^kt$, and there are 593 bacteria after 9 minutes, what is k?

0.4536 Signup and view all the answers

What is the simplified form of $(x^2 - 2x - 15)/(x^2 - x - 20)$ if possible?

x + 3 / x + 4 Signup and view all the answers

What is the simplified form of $6d^4/3e^3f × 15e^7f^2/20d^9$?

3e^4f/2d^5 Signup and view all the answers

What is the product of $(x - 5)(x + 2)/(x^2 - 25)$?

x + 2 / x + 5 Signup and view all the answers

What is the simplified result of $x^2 - 4 / x + 2 × 6x^2/2x - 4$?

3x^2 Signup and view all the answers

What is the solution for the equation $(x^2 + 9x + 20)/(x^2 - 25) ÷ (x + 4)/(x - 4)$?

x - 4 / x - 5 Signup and view all the answers

What is the result of adding $-2x - 6 / 9x + x + 6 / 9x$?

-1/9 Signup and view all the answers

What is the result of adding $5/x + 4 + 2/x - 4$?

7x - 12 / (x + 4)(x - 4) Signup and view all the answers

What is the simplified form of $5/x + 5 / 1/x - 6$?

-5x / (6x + 1)(x - 5) Signup and view all the answers

What are the solutions for the equation $2/x - 5 = x - 6$?

x = 4, x = 7 Signup and view all the answers

What is the solution for the equation $x^2/(x + 9) = 81/(x + 9)$?

x = ±9 Signup and view all the answers

What is the solution for the equation $x + 1/5x - 2/4x = 1/20$?

x = 2 Signup and view all the answers

What is the focus of the parabola given by $x = 1/32y^2$?

(8, 0) Signup and view all the answers

What is the equation in vertex form of the parabola with vertex at (0,0) and focus at (5,0)?

x = 1/20(y - 0) + 0 Signup and view all the answers

What conic section is represented by the equation $x^2 + 4y^2 = 4$?

ellipse Signup and view all the answers

Write the equation of an ellipse, in standard form, with vertices at (-7,0) and (7,0) and co-vertices at (0,-4) and (0,4).

(x - 0)^2/49 - (y - 0)^2/16 = 1 Signup and view all the answers

What conic section is represented by the equation $x^2/25 - y^2/9 = 1$?

hyperbola Signup and view all the answers

Find the coordinates of the center and radius for the circle defined by $x^2 + y^2 - 4x + 4y - 17 = 0$.

center: (2, -2) Signup and view all the answers

What are the solutions for the equation $x^2 - 12x = 28$?

x = 14, x = -2 Signup and view all the answers

What is the value of $i^{51}$?

i^{51} Signup and view all the answers

Given $f(x) = x^3 + 4x^2 + 2x - 3$, find $f(-20)$.

f(-20) = -6443 Signup and view all the answers

What is the simplified form of $(3x^2 - 12)/(4x^2 + 8x) / (6x - 12)/(8x^2 + 16x)$?

x + 2 Signup and view all the answers

Find the product of $(8 - 2i)(5 + i)$.

40 - 2t - 2t^2 Signup and view all the answers

Solve for the equation $log_2(x) + log_2(x-2) = log_2(3)$.

x = 3, x = -1 Signup and view all the answers

Solve $4^3x^{-1} = 1/256$.

x = -1 Signup and view all the answers

Simplify the expression $(3x^2/4y)^{-3}$.

64y^3/27x^6 Signup and view all the answers

How many positive zeroes does the polynomial $5x^2 - 3x - 2$ have?

1 positive Signup and view all the answers

What are the positive rational zeroes for $5x^2 - 3x - 2$?

2 Signup and view all the answers

What are the zeroes of the polynomial $5x^2 - 3x - 2$?

x = 1 Signup and view all the answers

Write in logarithmic form: $3^3 = 27$.

log_3(27) = 3 Signup and view all the answers

What is the domain of a polynomial function?

all real numbers Signup and view all the answers

Given the restricted domain for the polynomial function $f(x) = 2x^3 - 3x^2 - 2x + 1$ as $-3≤x≤3$, what is the range of the function?

range -86≤f(x)≤34 Signup and view all the answers

Which expression is equivalent to $3log(4) + log(6) - log(48)$ when written as a single log expression?

log(8) Signup and view all the answers

Solve the equation $2^x + 1 - 8^x - 2$.

x = 7/2 Signup and view all the answers

Solve the equation $ln(2x - 1) = 2$.

x = 1 + e^2/2 Signup and view all the answers

Solve for $x$ in the equation $-2x^3 + 14x^2 - 24x + 20 = 0$.

5, 1+i, 1-i Signup and view all the answers

Find the equation of the polynomial with roots $5, -4, 1$.

x^3 - 2x^2 - 19x + 20 = 0 Signup and view all the answers

Find the roots of the polynomial equation $x^3 - 2x^2 + 10x + 136 = 0$.

x - 3 = ±5i, x = 3 ± 5i, x = 4 Signup and view all the answers

Rationalize the denominator of $rac{√3 - √6}{√3 + √6}$.

-3 + 2√2 Signup and view all the answers

What is the center of the conic given by $rac{(x-3)^2}{36} - rac{(y+2)^2}{81} = 1$?

center (3, -2) Signup and view all the answers

Solve for $x$ in the equation $-6x - 4x = 6 + 5x$.

x = -2/5 Signup and view all the answers

Solve for $x$ in the equation $-1/2 - 4x/4 = 1/2 - 2x$.

5/6 Signup and view all the answers

Solve for $x$ in the equation $8x^2 = 12x - 4$.

x = 1/2, x = 1 Signup and view all the answers

Solve for $x$ in the equation $-5x^2 - 2 = 14x$.

-7 ± √39/5 Signup and view all the answers

Solve the following system of equations: $-3x - 9y = 7$, $8x + 9y = -1$.

x = 6/5, y = -53/45 Signup and view all the answers

Solve for $x$ in the equation $-2 - 2x = 2 + 2x/5$.

x = -5/3 Signup and view all the answers

What is the slope of the line given by the equation $y = rac{3}{5}x - 7$?

m = 3/5 Signup and view all the answers

In the arithmetic series described by $a_n = a_1 + (n - 1)d$, what does the variable $n$ represent?

n = number of terms Signup and view all the answers

Given the arithmetic series $-3, -3/2, 0, 3/2,...$, what is the common difference?

arithmetic difference = 3/2 Signup and view all the answers

What is the value of $log_2(16)$?

4 Signup and view all the answers

Find the sum of the arithmetic sequence with $a_1 = 60$, $d = 4$, $a_n = 120$.

s = 1440 Signup and view all the answers

What is the 80th term of the arithmetic sequence: $-1, 1, 3,...$?

a_{80} = 157 Signup and view all the answers

Find the sum of the arithmetic sequence: $8, rac{33}{4}, rac{17}{2}, rac{35}{4},..., 50$.

s = 4901 Signup and view all the answers

Given the arithmetic series: $a_n = -3(1/2)$, what is the common ratio?

1/2 Signup and view all the answers

Find the 5th term for the geometric series given by $a_1 = 2$, $r = 3$.

a_5 = 162 Signup and view all the answers

For the series $∑_{k=1}^∞ 3(2/3)^k$, does this series converge and what is the sum?

Sum = 6 Signup and view all the answers

What is the condition for an infinite geometric series to converge?

Common ratio < 1 Signup and view all the answers

Study Notes

Polynomial Roots and Complex Numbers

Real and Complex Roots: The equation (2x^3 + 4x^4 + 11x - 17 = 0) has one real root at (x=1) and complex roots given by (-\frac{3}{2} \pm \frac{5i}{2}).
Types of Solutions: The polynomial (x^5 + 2x^4 + 2x^3 = 0) has three real solutions (including multiplicity) at (x=0) and two imaginary solutions.

Simplification Techniques

Expression Simplification: Simplifying (6(4x+8y)+2(3x-6y)) yields (30x + 36y).
Radicals and Square Roots: The expression (10\sqrt{162} - \sqrt{405} - \sqrt{200}) simplifies to (80\sqrt{2} - 9\sqrt{5}).

Solving Equations

Absolute Value: The equation (|x-2| = 27) leads to two solutions: (x = 29) and (x = -25).
System of Equations: Solving (2x + 5y = 12) and (x - 3y = -2) gives (x = \frac{26}{11}) and (y = \frac{16}{11}).

Factoring Polynomials

Quadratic Factoring: The expression (4xy - 3y - 40x + 30) factors to ((4x-3)(y-10)).
Quadratic Expression: The equation (2x^2 + 3x - 9) factors to ((2x-3)(x+3)).

Operations with Complex Numbers

Complex Addition: Adding complex numbers ((5+i)-(6-12i)) results in (-1 + 13i).

Rational Expressions

Simplifying Expressions: (\frac{x^2 - 2x - 15}{x^2 - x - 20}) simplifies to (\frac{x+3}{x+4}).
Multiplying Rational Expressions: (\frac{x-5}{x^2-25} \cdot \frac{6x^2}{2x-4}) results in (3x^2).

Logarithmic Operations

Logarithm Evaluation: ( \log_2(8) ) evaluates to (x=3).
Single Log Expression: Combining logarithms, (\log_r(12) + \log_r(45)) simplifies to (\log_r(540)).

Conic Sections and Parabola Characteristics

Circle Characteristics: The circle (x^2 + y^2 - 4x + 4y - 17 = 0) has a center at ((2, -2)).
Parabola Focus: For the parabola (x=\frac{1}{32}y^2), the focus is at ((8, 0)).

Arithmetic and Geometric Sequences

Arithmetic Series Identification: The series (-3, -\frac{3}{2}, 0, \frac{3}{2},...) has a common difference of (\frac{3}{2}).
Geometric Series Sum: An infinite geometric series given by (\sum_{k=1}^{\infty} 3\left(\frac{2}{3}\right)^k) converges to (6).

Rationalizing Denominators

Rationalization Result: The expression (\frac{\sqrt{3}-\sqrt{6}}{\sqrt{3}+\sqrt{6}}) rationalizes to (-3 + 2\sqrt{2}).

Solving Polynomial Equations

Roots of Polynomial: The equation (-2x^3 + 14x^2 - 24x + 20 = 0) yields roots (5, 1+i, 1-i).
Polynomial Equation: The polynomial with roots (5, -4, 1) is given by (x^3 - 2x^2 - 19x + 20 = 0).

General Knowledge

Domain of Polynomial Functions: The domain of a polynomial function includes all real numbers.
Simple Linear Equations: In linear equations like (-6x - 4x = 6 + 5x), the solution is (x = -\frac{2}{5}).

These notes encapsulate key mathematical concepts and problem-solving techniques relevant to Algebra 2, providing a reference guide for key theorems and solutions.

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Test your knowledge with these Algebra 2 semester 2 exam flashcards. This quiz covers complex roots, simplifications, and solving equations, providing a comprehensive review of the material. Perfect for anyone preparing for exams in Algebra 2.