Advanced Algebra Practice Questions and Problems With Answers

‍

Brush up your skills or learn something new by going over some advanced algebra questions. 

‍

Mathematics is essential in understanding the world around us, from predicting the trajectory of a rocket to analyzing financial trends. Fields like calculus, physics, and linear algebra form the backbone of many scientific and engineering disciplines. 

But when it comes to pushing the boundaries of innovation, advanced algebra takes center stage. This branch of mathematics powers modern cryptography and computer algorithms, driving technological progress. 

Yet, mastering these concepts can be tough. That's why practicing with questions and problems is crucial. This blog will provide just that—a helpful resource for tackling challenging math head-on and making a meaningful impact in the world. 

What Is Advanced Algebra?

Advanced algebra is a branch of mathematics that deals with more complex algebraic concepts and techniques beyond the basic operations of arithmetic and elementary algebra. 

It typically includes topics such as polynomial equations and functions, systems of equations, inequalities, matrices, complex numbers, sequences and series, logarithms, exponentials, and graphing techniques.

Advanced algebra is fundamental in many areas of mathematics, science, engineering, and economics, providing tools for modeling, analyzing, and solving a wide range of problems.

How Is Advanced Algebra Applied in Practical Situations?

Advanced algebra is used in engineering, economics, science, and other areas to model complex systems, predict trends, and make informed decisions, helping drive progress in various fields. Let’s take a closer look. 

• Economic Modeling: Advanced algebra is vital in economics for predicting financial trends and assessing investment risks. Economists use algebra to understand market behaviors and make informed decisions about investments.
• Scientific Research: Scientists rely on advanced algebra to explore everything from tiny particles to vast celestial bodies. Algebraic equations help describe and analyze complex systems, driving discoveries and advancements in various scientific fields.
• Computer Science Algorithms: In computer science, advanced algebra is the backbone of algorithm development. By applying algebraic principles, computer scientists create efficient algorithms for tasks like internet searches and artificial intelligence, shaping our digital world.
• Statistical Analysis: In statistics, advanced algebra helps analysts uncover insights from large datasets. Techniques like regression analysis, based on algebra, reveal patterns and trends that inform decision-making across industries.
• Medical Imaging and Diagnosis: Doctors use advanced algebra in medical imaging technologies such as MRI scans. Algebraic concepts help visualize internal structures, aiding accurate diagnosis and treatment planning for patients.
• Environmental Modeling: Environmental scientists use advanced algebra to understand and protect ecosystems. Algebraic equations help model and predict the effects of climate change, guiding conservation efforts to preserve our planet's natural resources.

Advanced algebra is a tool used across different fields to solve problems, make decisions, and drive progress in our everyday lives.

Most Common Advanced Algebra Questions and Problems

1. Function g is defined by g(x) = 3(x + 8). What is the value of g(12)? 

A. –4 

B. 20 

C. 44 

D. 60

2. Which of the following is an equation of the line that passes through the point (0, 0) and is perpendicular to the line shown above? 

A. y = 54x 

B. y = 54 x + 3 

C. y = − 45 x 

D. y = − 45 x + 3

3. The surface area of a right rectangular prism can be found by finding the sum of the area of each of the faces of the prism. What is the surface area of a right rectangular prism with length 4 centimeters (cm), width 9 cm, and height 3 cm? (Area of a rectangle is equal to length times width.) 

A. 75 cm2

B. 108 cm2 

C. 120 cm2 

D. 150 cm2

4. Which of the following expressions is equivalent to (x + 7)(x2 – 3x + 2)?

A. x3 – 3x2 + 2x + 14 

B. x3 + 4x2 – 19x + 14 

C. x3 – 3x + 14 

D. x3 – 2x + 9

5. The graph below shows the cost, in dollars, of apples as a function of the number of pounds of apples purchased at a particular grocery store. The equation above defines the cost C, in dollars, for p pounds of pears at the same store. Which of the following statements accurately compares the cost per pound of apples and the cost per pound of pears at this store? 

A. Apples cost approximately $0.07 less per pound than pears do. 

B. Apples cost approximately $0.04 less per pound than pears do. 

C. Apples cost approximately $0.73 less per pound than pears do. 

D. Apples cost approximately $0.62 more per pound than pears do.

6. Which of the following is the graph of a function where y = f(x)?

7. Which of the following expressions is equivalent to 3x2 + 6x – 24? 

A. 3(x + 2)(x – 4) 

B. 3(x – 2)(x + 4) 

C. (x + 6)(x – 12) 

D. (x – 6)(x + 12)

8. A biologist puts an initial population of 500 bacteria into a growth plate. The population is expected to double every 4 hours. Which of the following equations gives the expected number of bacteria, n, after x days? (24 hours = 1 day) 

A. n = 500(2)x 

B. n = 500(2)6x 

C. n = 500(6)x 

D. n = 500(6)2x

9. x2+ 5x – 9 = 5 Which of the following values of x satisfies the equation above? 

A. 7 

B. 3 

C. –2 

D. –7 

10. The graph of y = f(x) is shown in the xy-plane below.

Which of the following equations could define f(x)? 

A.f(x)=x2 -2x-8 

B. f(x)=x2 +2x-8 

C. f(x) = (x – 2)(x + 4) 

D. f(x) = –(x – 1)2 – 9 

11. Which of the following best describes the range of y=-2x4+ 7? 

A. y ≤ –2 

B. y ≥ 7 

C. y ≤ 7 

D. All real numbers

12. For which of the following equations is x = 6 the only solution? 

A. (6x)2= 0 

B. (x – 6)2 = 0 

C. (x + 6)2 = 0 

D. (x – 6)(x + 6) = 0 

13. If f(x) = x2 + 3x + 1, what is f(x + 2)? 

A. x2 + 3x + 3 

B. (x + 2)2 + 3(x + 2) + 1 

C. (x + 2)(x2 + 3x + 1) 

D. x2 + 3x + 9 

14. What, if any, is a real solution to 5x +1 + 9 = 3? 

A. −15 

B. 7 

C. 1435

D. There is no real solution. 

15. If x ≠ –2 and x ≠ 32, what is the solution to 5x + 2 = x 2x − 3 ? 

A. 3 and 5 

B. 2 and − 32 

C. –2 and 32 

D. –3 and –5

16. Triangle JKL and triangle PQR are shown above. If ∠J is congruent to ∠P, which of the following must be true in order to prove that triangles JKL and PQR are congruent? 

A. ∠L ≅ ∠R and JL = PR 

B. KL = QR and PR = JL  

C. JK = PQ and KL = QR 

D. ∠K ≅ ∠Q and ∠L ≅ ∠R

17. In the function f(x) = a(x + 2)(x – 3)b , a and b are both integer constants and b is positive. If the end behavior of the graph of y = f(x) is positive for both very large negative values of x and very large positive values of x, what is true about a and b? 

A. a is negative, and b is even. 

B. a is positive, and b is even. 

C. a is negative, and b is odd. 

D. a is positive, and b is odd.

FAQs

Let’s take a look at some commonly asked questions surrounding advanced algebra. 

1. What Concepts Are Covered in Advanced Algebra?

In advanced algebra, you'll learn about polynomial equations, systems of equations, inequalities, matrices, complex numbers, sequences, series, logarithms, exponentials, and graphing. 

2. Can You Provide an Example of a Hard Advanced Algebra Question?

A challenging question in advanced algebra involves the sum of cubes problem, represented by the equation x3+y3+z3=k. 

When approached as a Diophantine equation, where x, y and z must be whole numbers for any given value of k, finding solutions becomes quite tricky. For instance, solving for a specific k value, such as k=33, requires finding integer solutions for x, y and z, which can be quite challenging and typically involves advanced algebraic techniques and extensive calculations.

3. How to Solve Advanced Algebra Easily?

To solve advanced algebra problems easily, master basic concepts, practice regularly, and use problem-solving techniques like factoring. Seek help when needed, break down complex problems, and stay organized. Review solutions, learn from mistakes, and stay positive and persistent in your studies.

Final Thoughts

Mastering advanced algebra opens doors to understanding complex systems in economics, science, and technology. Practice makes perfect, so work through these questions to sharpen your skills. Keep pushing forward, and let advanced algebra be your passport to new discoveries and opportunities.

‍