Overview and Formulas Of Algebra Concepts Tested On GRE

The GRE examination also comprises algebraic concept tests wherein students are more likely to be tested on their basic math skills, reasoning, and even understanding elementary mathematical concepts. This is one of the GRE Quantitative Reasoning sections, and these concepts are important to allow the examinees to answer the test questions related to this subject.

Is algebra in the GRE Quantitative Section?

Yes, algebra is in the GRE Quantitative Section. The examination, which includes algebra, basically assesses the candidates using their knowledge of solving operations with exponents, relations, functions, and inequalities; solving word problems; coordinating geometry; or even intercepts and slopes of lines.

What are the algebra concepts tested on the GRE?

The algebra concepts tested on the GRE are the following:

  • Simplifying Algebraic Expressions. Through this concept, it simply has the objective of defining a process of solving in the most efficient and compact form without literally changing the whole value.
  • Expanding Expressions. Aside from simplifying, there is also a process wherein expanding expressions is needed. Expanding algebraic expressions is more likely to involve combining more than one variable or number using the algebraic expression given to remove any brackets or parentheses by combining like terms.
  • 3 types of factoring. It is basically used to break down the numbers, or factors, that multiply together to form another number.
  • Basic Equations. In this coverage, basic algebraic equations cover simple operations of mathematics just like addition, multiplication, subtraction, or division, which might involve constants or variables.
  • Systems of Equations. In this part of the question, a collection of two or more equations with the same set of unknowns is introduced, and we will be trying to find the values of those unknowns that will satisfy every equation.
  • Quadratic Equations. In this concept, a quadratic equation is any equation that can be rearranged into a standard form wherein x is the unknown value, and the rest are known numbers used together with the formula.
  • Equations with exponents. One of the questions that the examinees could face on this examination will be exponential equations. In this area, the variable occurs as an exponent, and you need to have equations with comparable exponential expressions.
  • Equations with Fractions. It is also possible that examinees could be asked questions that consist of fractions or a representation of equal parts of a whole or a collection. It may require solving equations by clearing the denominators, or even finding the least common denominator of all fractions.
  • Equations with square roots. The square is used to keep negative numbers from reeking into chaos. It is useful to allow a continuous direction of lines from one point to another.
  • Equations with absolute values. Basically, the whole definition of this category is to define the absolute difference between arbitrary real numbers and the standard metric on those real numbers.
  • The Coordinate Plane. The coordinate plane is one of the possible questions present on the exam. The examinees may be presented with graphs or a set of values that show an exact position.
  • Equations of lines. These equations make an easy way to plot and compare two or more lines to each other, wherein they can form a slope or coordinate plane.
  • Graphs of quadratics. The graph of a quadratic function is a U-shaped curve called a parabola. It is one of the possible topics covered in the GRE exam where the candidates will possibly be asked to answer.

During the examination proper, examinees will be asked about either of these concepts. That is for them to be assessed on their knowledge, which comprises all the possible coverage of Algebra. Generally, the questions asked are most probably the basic algebraic equations up to complex ones. That's why it is important to be well familiarized and proficient with these topics.

What are the fundamental algebraic expressions required for the GRE?

The basic algebraic expressions needed in the GRE are listed below.

  • Combining Like Terms. It is one of the processes wherein an expression is simplified using addition or subtraction terms from the same variable factors.
  • Adding and Subtracting Polynomials. It is used to simply add or subtract polynomials by combining like terms.
  • Factoring Algebraic Expressions. Factoring is one way of factoring polynomials meant to express a product or two for simpler expressions. It can be done by using a distribution factor.

The following list above contains the necessary terms, operations, and/or expressions that are needed to know on the GRE examination. These basic algebraic expressions are most likely to be asked on the exam.

What are the advanced algebraic expressions needed on the GRE?

The advanced algebraic expressions test takers must know on the GRE are as follows:

  • Substitution. This is used to evaluate algebraic expressions or express them in terms of other variables. It simply replaces variables in the expression with the number or quantity specified for its equivalent while employing the specified operations and order, as in PEMDAS.
  • Solving for One Unknown in Terms of Another. Some GRE problems do not require you to solve for the numerical value of an unknown. However, some may ask you to solve for the variable of one in terms of the other. To do so requires transposing all constants and other variables to the other side.
  • Simultaneous Equations. For this part, obtaining enough information to compare two quantities usually requires as many equations as your variables. Those equations will give you different information about the same two variables.
  • Symbolism. On the GRE examination, there is a possibility that you will see strange symbols like stars. No need to worry, because this type usually requires nothing more than substitution.
  • Sequences. These are lists of numbers and in the GRE, they could be represented as this: s s₂, s₃,…sn,.. The subscript part gives you the position of each element in the series.

This list is merely one of the possible contents that an examinee could face during the examination. These are advanced algebra expressions used for the examination, and by knowing these, it will probably be a smooth sailing exam for those who are planning to take it.

List of GRE Algebraic Sample Questions

The GRE Algebraic question samples can be found below:

Example 1: What is the value of t if: 3x2 + tx - 21 = (3x - 3)(x + 7)?

Possible Answers:

  • 18
  • 21
  • -18
  • -3
  • 24

Correct answer: 18

Explanation:

Use the foil method: (3x - 3) (x + 7) = 3x2 +21x - 3x - 21 = 3x2 +18x -21 so t = 18.

  1. Expand the following equation:

(x3−3)(x+7)

Possible Answers:

  • x4−4x−21
  • x2+14x−21
  • x2+4x+21
  • x4+7x3−3x−21
  • x2−21

Correct answer:

x4+7x3−3x−21

Explanation:

Use FOIL to factor the expression.

First: (x3)(x) = x4

Outside (x3)(7) = 7x3

Inside (–3)(x) = –3x (Don't forget the negatives!)

Last (3)(7) = –21

Example 2: Quantitative Comparison

Quantity A: 22+32

Quantity B: (2+3)2

Possible Answers:

  • Quantity B is greater.
  • Quantity A is greater.
  • The relationship cannot be determined from the information given.
  • The two quantities are equal.

Correct answer:

Quantity B is greater.

Explanation:

Quantity A: 22 + 32 = 4 + 9 = 13

Quantity B: (2 + 3)2 = 52 = 25, so Quantity B is greater.

We can also think of this in more general terms. x2 + y2 does not generally equal (x + y)2.

Example 3: ( x + 3y ) ( x - 3y ) = 8

Quantity A: x2−9y2

Quantity B: 16

Possible Answers:

  • The two quantities are equal.
  • Quantity A is greater.
  • The relationship cannot be determined from the information given.
  • Quantity B is greater.

Correct answer:

Quantity B is greater.

Explanation:

The difference of squares formula says (x + a)(x - a) = x2 - a2.

Thus, Quantity A equals 8.

Therefore, Quantity B is greater.

Example 4: x < 0 | y < 0

Quantity A: (x+y)2

Quantity B: x2+4xy+y2

Possible Answers:

  • Quantity A is greater.
  • The relationship cannot be determined.
  • Quantity B is greater.
  • The two quantities are equal.

Correct answer:

Quantity B is greater.

Explanation:

To approach this problem, consider the two quantities

Quantity A: (x+y)2

Quantity B: x2+4xy+y2

They are in different forms, so expand quantity A:

Quantity A: x2+2xy+y2

Quantity B: x2+4xy+y2

Now, for the purpose of comparison, subtract shared terms from each quantity:

Quantity A*: 0

Quantity B*: 2xy

Both x and y are negative, non-zero values. Since 2xy is a product of two negative values, it must be positive. Quantity B must be greater than Quantity A.

Example 5: Expand the function: (xy3+x2y)(xy−x3y2)

Possible Answers:

−x5y3−x4y5−x3y2−x2y4

x5y3+x4y5+x3y2+x2y4

−x5y3−x4y5−x3y2+x2y4

−x5y3−x4y5+x3y2+x2y4

x5y3−x4y5+x3y2+x2y4

Correct answer:

−x5y3−x4y5+x3y2+x2y4

Explanation:

Use the method of FOIL (First, Outside, Inside, Last) and add exponents for like bases:

(xy3+x2y)(xy−x3y2)

xy3(xy)−xy3(x3y2)+x2y(xy)−x2y(x3y2)

x2y4−x4y5+x3y2−x5y3

−x5y3−x4y5+x3y2+x2y4

Example 6: x < 0 | y > 0

Quantity A: (x+y)3

Quantity B: x3+y3

Possible Answers:

  • Quantity B is greater.
  • Quantity A is greater.
  • The relationship cannot be determined.
  • The two quantities are equal.

Correct answer:

The relationship cannot be determined.

Explanation:

Begin by expanding Quantity A:

(x+y)3

x3+3x2y+3xy2+y3

Now in order to compare this to Quantity B:

x3+y3

A good method would be to subtract shared terms from each Quantity; in this case, both quantities have an x3 and y3 term. Removing them gives:

Quantity A' : 3x2y+3xy2

Quantity B' : 0

The question now is the sign of Quantity A'; if it's always positive, Quantity A is greater. If it's always negative, Quantity B is greater. If it is zero, the two are the same.

We only know that

x<0

y>0

If x=−1;y=1, then Quantity A' would be zero.

If x=−2;y=1, then Quantity A' would be positive.

Since values of x and y can be chosen to vary the relationship, the relationship cannot be determined.

Example 7: x < 0, y > |x|

Quantity A: (x+y)3

Quantity B: x3+y3

Possible Answers:

  • The two quantities are equal.
  • Quantity B is greater.
  • Quantity A is greater.
  • The relationship cannot be determined.

Correct answer:

Quantity B is greater.

Explanation:

Begin by expanding Quantity A:

(x+y)3

x3+3x2y+3xy2+y3

Now in order to compare this to Quantity B:

x3+y3

A good method would be to subtract shared terms from each Quantity; in this case, both quantities have an x3 and y3 term. Removing them gives:

Quantity A' : 3x2y+3xy2

Quantity B' : 0

The question now is the sign of Quantity A'; if it's always positive, Quantity A is greater. If it's always negative, Quantity B is greater. If it is zero, the two are the same.

We know that

x<0

y>|x|

Now compare 3x2y and 3xy2:

Looking at absolute values so that we're only considering positive terms:

|3xy|

y>|x|

From this it follows that by multiplying |3xy| across the inequality :

|3xy2|>|3x2y|

From this we can determine that the magnitude of |3xy2| is greater. However, since this is the product of one negative number and two positive numbers, 3xy2 is negative, and the sum of 3xy2 and 3x2y must in turn be negative, and so Quantity A' must be negative!

Example 8:

Quantity A: x2+5x−14x−2

Quantity B: x+7

Possible Answers:

  • Quantity A is greater.
  • Quantity B is greater.
  • The two quantities are equal.
  • The relationship cannot be determined.

Correct answer:

The relationship cannot be determined.

Explanation:

This problem is deceptive. Looking at Quantity A, one may think to factor and reduce it as follows:

x2+5x−14x−2

(x+7)(x−2)x−2

x+7

Which is identical to Quantity B.

However, we cannot ignore that x−2 in the original fraction! We are given no conditions as to the value of x. If x=2, then Quantity A would be undefined. Since we're not given the condition x≠2, we cannot ignore this possibility.

The relationship cannot be established.

Example 9: Solve the following expression, (x−2)2.

Possible Answers:

  • x2−4x−4
  • x2−2
  • x2+4x+4
  • x2+4
  • x2−4x+4

Correct answer:

x2−4x+4

Explanation:

You must FOIL the expression which means to multiply the first terms together followed by the outer terms, then the inner terms and lastly, the last terms.

The expression written out looks like

(x−2)(x−2).

You multiple both First terms to get x2.

Then the outer terms are multiplied (x∗−2=−2x).

Then you multiple the inner terms together (−2∗x=−2x).

Finally, you multiply the last terms of each (−2∗−2=4).

This gives you x2−2x−2x+4 or x2−4x+4.

The following examples that are listed above comprise basic and advanced algebraic questions and equations. Those examples are mainly one of the coverages that is asked during the exam wherein examinees are expected to give the answers based on the list of possible answers on the test.

What is the solving overview of algebraic operations and simplifying expressions?

The following are the basic rules and steps for simplifying any algebraic expression:

  • Remove any grouping symbols such as brackets and parentheses by multiplying factors.
  • Use the exponent rule to remove groupings if the terms contain exponents.
  • Combine like terms by addition or subtraction.
  • Combine the constants.

Solving this type of question during the exam will mainly ask you to write an expression in the most efficient and compact form while still maintaining the value of the original expression.

What is the solving overview of the Rules of Exponents and Radicals?

The solving overview of the Rules of Exponents and Radicals can be found below:

  • Isolate the radical expression on one side of the equal sign. Put all the remaining terms on the other side.
  • If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for the nth root radical, raise both sides to the nth power. Doing so eliminates the radical symbol.
  • Solve the resulting equation.
  • If a radical term still remains, repeat steps 1–2.
  • Substitute solutions by substituting them into the original equation.

A common rule, aside from the mentioned solutions, is that raising a negative exponent means making the reciprocal positive.

What is the solving overview of Solving Equations in One Variable?

Solving equations in one variable can be resolved using LCM. Clear the fractions, if any. Then, simplify both sides of the equation. After that, Isolate the variable. Lastly, test takers must verify their answer.

Example 1 : Solve for x, 2x – 4 = 0

Solution:

Add 4 both sides

2x – 4 + 4 = 0 + 4

2x = 4

Divide each side by 2, we get

2x/2 = 4/2

x = 4/2 = 2

So, x = 2 is the answer.

What is the solving overview of Linear/Quadratic Inequalities?

With quadratic equations, always begin by getting it into standard form:

Ax2+Bx+C=0

Therefore, take our equation:

3x2−5x=2x+6

And rewrite it as:

3x2−7x−6=0

You could use the quadratic formula to solve this problem. However, it is possible to factor this if you are careful. Factored, the equation can be rewritten as:

(3x+2)(x−3)=0

Now, either one of the groups on the left could be 0 and the whole equation would be 0. Therefore, you set up each as a separate equation and solve for x:

3x+2=0

3x=−2

x=−23

OR

x−3=0

x=3

The sum of these values is: 3+(−23)=3−23=93−23=73

What is the solution overview for Solving Simultaneous Equations in Two Variables?

In this area, there are three common methods for solving it that are used to find its answer. This could be addition and subtraction, substitution, and graphing.

  • Addition/subtraction method. This method is also known as the "elimination method."
  1. Multiply one or both equations by some number(s) to make the number in front of one of the letters (unknowns) in each equation the same or exactly the opposite.
  2. Add or subtract the two equations to eliminate one letter.
  3. Solve the remaining unknown.
  4. Solve for the other unknown by inserting the value of the unknown found in one of the original equations.
  • Substitution method. This is a method that involves substituting one equation for another.
  • Graphing method. This method involves solving equations using representations of graphs. Those coordinates placed on the graph will be the solution to the system.

What is a summary of word problems and applications that explains how to solve them?

The solving overview when it comes to word problems and applications is the following:

  1. Identify what is being asked for in the problem.
  2. Pull out all the information given from the selection.
  3. If you can, think of an equation that you can use to solve the problem.
  4. Review all the necessary information that will be used for your computation.
  5. Solve the equation.
  6. Make sure that you have given an answer to what is being asked.

This type of question that will be asked during the GRE examination requires understanding and good analysis. You have to be keen enough to distinguish between what is being asked and the necessary things for you to answer.

What is the solving overview of functions and their graphs?

The vertical line test can be used to determine if an equation is a function. To be considered a function, there must be only one y (or f(x)) value for each x value. The vertical line test determines how many y (or f (x)) values are present for each value of x. If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function. The horizontal line test can be used to determine if a function is one-to-one, that is, if only one x value exists for each y (or f (x)) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

What is the solving overview of Intercepts, Slope, and Other Topics in Coordinate Geometry?

What is the y-intercept of the line that goes through the points (–2, 1) and (5, 6)?

Possible Answers:

17/7

–2/7

0

67/7

The answer cannot be determined from the given information.

Correct answer:

17/7

Explanation:

The slope can be calculated from m = (y2 – y1)/(x2 – x1) = (6 – 1)/(5 + 2). Having calculated the slope, we can now use point-slope form of a line, y – y1 = m(x – x1), and using the second point (5, 6): y – 6 = (5/7)(x – 5). This can be rearranged into slope-intercept form to obtain: y = (5/7)x + (17/7). Because the equation is now in slope intercept form, we know that the y-intercept is 17/7.

What are some algebra quantitative test-taking tips?

Here are some test-taking tips for the actual GRE Mathematics section:

  1. Before answering them, visualize the GRE questions. This technique could really help you seek an approach to how you solve the question being asked. For the GRE examination, understanding and good analysis are required for you to provide a correct answer.
  2. Spend less time on one question. Taking your time to read is a great way for you to analyze the questions well. However, taking too much time for a single question can have a consequence or the possibility of not answering the other problems.
  3. Don’t be pressured and keep calm during the test. Before the test starts, try to put your mind and your body at ease. With this, you will be able to think clearly and minimize all those unnecessary thoughts that are not beneficial to you.
  4. Eliminate what's not necessary. During the test, given that there were a lot of choices provided, some of them would really trick you into choosing the correct or wrong answer. One of the best ways to avoid it is to employ elimination. This strategy will allow you to choose the best answer while avoiding the wrong ones.
  5. Make sure you're careful when you're writing your response. Always choose to review your answer well because some wrong choices can look similar to the correct ones sometimes.
  6. All of the questions must be answered. Never leave a single question unanswered. Instead of leaving that certain number unanswered, it is better to attempt it because there could be a possibility wherein you can get a point with that.
  7. Pay close attention to the instructions. Follow all the instructions being given for the exam. This will prevent you from making mistakes, and could even prevent you from getting negative scores.
  8. Familiarize yourself with the format of the GRE exam. Being able to familiarize yourself with the test format will prevent you from being surprised by the type of questions you will be facing. Learning all the different types of questions will most likely help you understand them and allow you to solve them well.
  9. Create a consistent study plan. There is no better way than preparing yourself and giving yourself time to practice all the possible questions. With this preparation, you are more likely to give yourself some time to comprehend basic to advanced problems.
  10. Solve practice tests in a timed setting. This strategy is good practice when it comes to taking an examination with a timed setting. It will allow you to get used to this type of testing condition and allow you to manage your time properly.

These are some of the tips that are really helpful for your algebra quantitative test. Some of the questions and answers are already tricky, so it is better to exercise yourself and get that high score that you are aiming to achieve for this test. (5) formalized paraphrase

References:

  1. https://www.kaptest.com/study/gre/whats-tested-on-the-gre-algebra/
  2. https://gre.economist.com/gre-advice/gre-quantitative-section/algebra-gre/algebra-concepts-tested-gre-overview
  3. https://www.prepscholar.com/gre/blog/gre-math-review-topics/
  4. https://www.varsitytutors.com/gre_math-help/algebra
  5. https://crunchprep.com/gre/essential-gre-math-tips