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Quizzes > Mathematics & Statistics

Algebra Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art illustrating the concept of Algebra course

Boost your algebra skills with our engaging Algebra practice quiz designed to reinforce core concepts like factoring, rational expressions, equations and inequalities, and functions and graphs. Dive into essential topics such as exponential and logarithmic functions, systems of equations, matrices, determinants, and the binomial theorem - perfect for a rapid review and building confidence ahead of your next challenge.

Factor the quadratic expression x² - 5x + 6.
(x - 2)(x + 3)
(x - 1)(x - 6)
(x - 2)(x - 3)
(x + 2)(x + 3)
The factors of 6 that add to -5 are -2 and -3, which factor the quadratic as (x - 2)(x - 3). This technique is a key method in factoring quadratics.
Factor the expression x² - 16.
(x - 4)(x + 4)
(x - 16)(x + 1)
(x - 2)(x + 8)
(x - 8)(x + 2)
The expression is a difference of squares which factors as (x - 4)(x + 4) because 16 is 4². Recognizing these patterns is essential in algebra.
Simplify the rational expression (x² - 9)/(x + 3).
x + 3
1/(x - 3)
x² - 3
x - 3
The numerator factors as (x - 3)(x + 3) and the common factor cancels with the denominator, giving x - 3. This process highlights the importance of factoring in simplifying rational expressions.
Solve the linear equation 2x + 5 = 11.
2
-3
5
3
Subtracting 5 from both sides yields 2x = 6, and dividing by 2 gives x = 3. This straightforward method is essential for understanding algebraic manipulations.
What is the slope of the line represented by the equation y = 2x + 1?
2
1
1/2
-2
In the slope-intercept form y = mx + b, the coefficient m represents the slope. Here, m equals 2, so the slope of the line is 2.
Solve for x in the exponential equation 3^x = 81.
3
4
-4
2
Since 81 can be written as 3^4, we equate the exponents to obtain x = 4. This approach of matching bases is fundamental in solving exponential equations.
Solve for x if log₂(x) = 5.
16
32
64
8
The logarithmic equation log₂(x) = 5 implies that x = 2^5, which evaluates to 32. Understanding logarithms is essential for converting between exponential and logarithmic forms.
Solve the system of equations: 2x + y = 7 and x - y = 1.
(8/3, 5/3)
(2, 3)
(3, 1)
(1, 5)
By isolating y from the second equation and substituting into the first, the solution (8/3, 5/3) is obtained. This demonstrates effective use of the substitution method in solving linear systems.
Calculate the determinant of the matrix [[3, 5], [2, 4]].
2
8
-2
7
The determinant of a 2x2 matrix is computed as ad - bc. Substituting the given values yields 3×4 - 5×2 = 2.
Using the Remainder Theorem, find the remainder when 2x³ - 3x + 4 is divided by (x - 1).
2
3
-3
4
According to the Remainder Theorem, the remainder is P(1). Evaluating the polynomial at x = 1 gives 2 - 3 + 4 = 3.
Simplify the rational expression (x² - 4)/(x² - x - 6).
(x - 2)/(x - 3)
(x + 2)/(x - 2)
No simplification
(x - 3)/(x - 2)
Factoring yields x² - 4 = (x - 2)(x + 2) and x² - x - 6 = (x - 3)(x + 2). Canceling the common factor (x + 2) gives (x - 2)/(x - 3).
Solve the inequality 2x - 3 > 5.
x < 4
x > 4
x ≥ 4
x ≤ 4
Adding 3 to both sides gives 2x > 8 and dividing by 2 results in x > 4. Correctly manipulating the inequality is crucial for proper solution.
Factor completely: x³ - x² - 4x + 4.
(x + 1)(x - 2)(x + 2)
(x - 1)(x - 2)(x + 2)
(x - 1)(x² - 4)
(x - 1)(x - 2)
Grouping the terms yields x²(x - 1) - 4(x - 1) which factors to (x - 1)(x² - 4). Further factoring using the difference of squares produces (x - 1)(x - 2)(x + 2).
Using the Binomial Theorem, determine the coefficient of x² in the expansion of (1 + x)❵.
10
6
15
5
The Binomial Theorem tells us that the coefficient of x² in (1 + x)❵ is given by C(5, 2), which equals 10. This demonstrates the use of combinatorial principles in expanding binomials.
Identify the transformation that describes the graph of y = 3^(x - 2) + 4 compared to y = 3^x.
Shifted right by 2 and up by 4
Shifted left by 2 and down by 4
Shifted right by 2 and down by 4
Shifted left by 2 and up by 4
The expression (x - 2) in the exponent indicates a horizontal shift to the right by 2 units, while the +4 outside the exponent indicates a vertical shift upward by 4. Recognizing these transformations is key to graphing functions effectively.
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Study Outcomes

  1. Apply factoring techniques to simplify algebraic expressions.
  2. Solve equations and inequalities involving rational and polynomial expressions.
  3. Graph and analyze functions, including exponential and logarithm functions.
  4. Evaluate systems of equations using matrices and determinants.
  5. Apply the binomial theorem to expand polynomial expressions.

Algebra Additional Reading

Ready to dive into the world of algebra? Here are some top-notch resources to guide you through your mathematical journey:
  1. Algebra I by Prof. Michael Artin This MIT OpenCourseWare course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups, providing a solid foundation in algebra.
  2. Algebra I Student Notes These student-created notes from MIT offer insights into modern algebra topics, including groups, linear algebra, and geometry, complementing the formal course materials.
  3. Online Textbooks from UC Davis A collection of free online textbooks covering various mathematical topics, including linear algebra and differential equations, authored by UC Davis faculty.
  4. A First Course in Linear Algebra This study guide provides comprehensive coverage of linear algebra concepts, complete with exercises and solutions, ideal for reinforcing your understanding.
  5. Linear Algebra by Prof. Gilbert Strang An MIT OpenCourseWare course offering video lectures, problem sets, and exams to deepen your grasp of linear algebra concepts.
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