Graduate Record Examinations (GRE) Quantitative Reasoning: Algebra Syllabus

Quantitative Reasoning: Algebra syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Reasoning: Algebra in Graduate Record Examinations (GRE), not a summary of it.

1. 1

   Expressions and Equations

   3 topics

   • Simplifying algebraic expressions
     • Combining like terms
     • Factoring and expanding
   • Linear equations and systems
     • Substitution and elimination
     • Solutions and consistency
   • Quadratic equations
     • Factoring and the quadratic formula
     • Discriminant and nature of roots
2. 2

   Inequalities and Absolute Value

   3 topics

   • Solving linear inequalities
     • Sign-flip rules
     • Compound inequalities
   • Absolute value equations and inequalities
     • Distance interpretation
     • Case analysis
   • Quadratic and rational inequalities
     • Sign charts
3. 3

   Functions, Graphs and the Coordinate Plane

   3 topics

   • Function notation and evaluation
     • Domain and range
     • Composite functions
   • Lines in the coordinate plane
     • Slope and intercepts
     • Parallel and perpendicular lines
   • Graphs of common functions
     • Parabolas and transformations
     • Reading graphs and intersections
4. 4

   Word Problems and Modelling

   3 topics

   • Translating words to equations
     • Defining variables
   • Mixture, age and money problems
     • Setting up balance equations
   • Sequences and patterns
     • Arithmetic and geometric sequences

Quantitative Reasoning: Algebra flashcards for Graduate Record Examinations (GRE)

20 of 52 cards from the Quantitative Reasoning: Algebra deck — real questions with worked answers.

1. What is the distributive property used to expand $a(b+c)$?

   $a(b+c) = ab + ac$. Multiply the outside term by each term inside the parentheses.

2. How do you combine like terms in $5x + 3x^{2} - 2x + 7x^{2}$?

   Add coefficients of terms with identical variable parts: $5x - 2x = 3x$ and $3x^{2} + 7x^{2} = 10x^{2}$, giving $10x^{2} + 3x$.

3. State the three exponent laws for products, quotients, and powers of powers.

   $x^{a}\cdot x^{b} = x^{a+b}$, $\dfrac{x^{a}}{x^{b}} = x^{a-b}$, and $(x^{a})^{b} = x^{ab}$.

4. What does a negative exponent mean, e.g. $x^{-n}$?

   $x^{-n} = \dfrac{1}{x^{n}}$ for $x \neq 0$. A negative exponent is the reciprocal of the positive power.

5. What is the value of any nonzero base raised to the zero power, $x^{0}$?

   $x^{0} = 1$ for any $x \neq 0$.

6. How do you add the rational expressions $\dfrac{a}{b} + \dfrac{c}{d}$?

   Use a common denominator: $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$.

7. What are the standard steps to solve a linear equation in one variable?

   Simplify each side (distribute, combine like terms), move variable terms to one side and constants to the other, then divide by the coefficient of the variable.

8. Solve $3x - 7 = 11$ for $x$.

   $3x = 18$, so $x = 6$.

9. What are the three solution outcomes possible for a system of two linear equations in two variables?

   One unique solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same line).

10. Describe the substitution method for solving a linear system.

    Solve one equation for one variable, substitute that expression into the other equation, solve for the remaining variable, then back-substitute.

11. Describe the elimination (addition) method for solving a linear system.

    Scale equations so one variable's coefficients are opposites, add the equations to eliminate that variable, solve, then back-substitute.

12. Solve the system $x + y = 10$ and $x - y = 4$.

    Add the equations: $2x = 14$, so $x = 7$ and $y = 3$.

13. What is the standard form of a quadratic equation?

    $ax^{2} + bx + c = 0$ with $a \neq 0$.

14. State the quadratic formula.

    $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$ for $ax^{2} + bx + c = 0$.

15. What is the discriminant, and what does its sign tell you?

    The discriminant is $\Delta = b^{2} - 4ac$. If $\Delta > 0$ there are two real roots, if $\Delta = 0$ one repeated real root, if $\Delta < 0$ no real roots.

16. What is the zero-product property and how is it used to solve quadratics?

    If $pq = 0$ then $p = 0$ or $q = 0$. After factoring $ax^{2}+bx+c=0$ into factors, set each factor equal to zero to find the roots.

17. Factor $x^{2} - 5x + 6$.

    $(x-2)(x-3)$, since $-2$ and $-3$ multiply to $6$ and add to $-5$.

18. State the difference of squares factoring formula.

    $a^{2} - b^{2} = (a+b)(a-b)$.

19. What are the perfect-square trinomial expansions?

    $(a+b)^{2} = a^{2} + 2ab + b^{2}$ and $(a-b)^{2} = a^{2} - 2ab + b^{2}$.

20. By Vieta's formulas, what are the sum and product of the roots of $ax^{2}+bx+c=0$?

    Sum of roots $= -\dfrac{b}{a}$; product of roots $= \dfrac{c}{a}$.

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