Week 5, Day 2: Algebra Review
Chapter 12: Math Foundations – Algebra Review
Memorize:
·
Algebraic terms
o
Variable – letters representing unknown
quantities
o
Constants – numbers not multiplied by any
variable
o
Terms – constants and products of constants and
variables
o
Product – result of multiplication
o
Coefficient – the constant in front of a
variable; if no number is visible, the coefficient is understood to be 1
o
Algebraic expression – any mathematical
expression (‘sentence’) containing one or more constants, variables, and
possibly one or more operation symbols (e.g., +- etc.)
o
Monomial – algebraic expressions with a single
term (no spaces or operators, e.g., 2, 3a, 4xyz, etc.)
o
Polynomial – algebraic expressions with more
than one term (combined with operators and spaces, e.g., 2 + 3a, etc.)
§
Binomial – has exactly two terms
§
Trinomial – has exactly three terms
§
Etc.
o
Algebraic equation – two expressions set equal
to each other
·
Basic operations
o
Combining like terms – add or subtract terms
with the same variables (3a can be added to 2a, but not to 4b)
o
Adding and subtracting polynomials (combine like
terms)
o
Factoring – simplifying by pulling out greatest
common factors using the distributive law.
·
Order of Operations (PEMDAS) – parentheses,
exponents, multiplication, division, addition, subtraction
·
FOIL – first, outer, inner, last
·
Divisibility & Multiples
·
Even vs. Odd
·
See also Appendix D!
Advanced Operations
Substitution
Remember to follow PEMDAS, and replace every variable with the matching number it equals. Then solve the equation with the values given.
Solving Equations
Always do the same thing to both sides of the equal sign; without exponents, put all variables on one side and all constants on the other.
Inequalities
Just like solving any equation, EXCEPT when multiplying or dividing by a negative number: then you must flip the direction of the inequality sign.
Solving for One Unknown in Terms of Another
Isolate one variable to one side of the equal sign by doing PEMDAS in reverse order.
Simultaneous Equations (Systems of Equations)
REMEMBER: for any group of equations, you can only solve it IF the number of variables is EQUAL to the number of unique equations.
Two methods: substitution and combination
In substitution: solve one equation in terms of one
variable, and plug that into the other equation. You should be able to solve
for that variable, and then use the original equation to solve for the other
variable.
In combination: stack the equations vertically, then add or subtract them. You might have to multiply one equation by a number to get one of the variables to cancel out through addition or subtraction.
Symbolism
See a strange symbol? Don’t panic! It is just a function.
From Math 1: “False” Operations (a.k.a., Functions)
The GRE and GMAT both use false operations which include strange symbols set equal to some algebraic expression and then ask you to solve an operation with some given values. The strange symbols I’ve seen include: @ # $ * §¨©ª†♫«■●▲ and other strange shapes I cannot even find in the symbol library.
All of these have the same meaning as f(x) or g(x); that is to say, they are all just functions. So if you see x @ y = x ´ 3y, and you want to solve for 3 @ 4, then you simply plug 3 in for x, and 4 in for y, and then solve.
Remember that functions can nest together like f(g(x)) or f ° g, so you might see a pair of symbolic functions where you have to solve them from the inside out. We will cover these in the next level of GRE/GMAT Math.
Remember: The key is to substitute the number value for the variable letter in the equation, and then solve. This is just the same plug’n’play we do with any expression and a given numerical quantity for a variable.
Sequences
A sequence is just a list of numbers, which we can call by a
letter variable with a subscript numeral to track the order of the sequence.
For example, s1, s2, s3,…sn (“s sub
1, s sub 2, s sub 3, s sub n”) could equal 1, 2, 3, …etc. The key to these is
simply discovering the pattern and following the instructions.
Polynomials and Quadratics
FOIL
“First, Outer, Inner, Last” means that given the product of
two binomials like (x + y)(x + y), we multiply first term to first term (x
times x), outer to outer (x times y), inner to inner (y times x) and last to
last (y times y).
See: https://gregmatmath1.blogspot.com/2020/05/week-3-day-2.html
Factoring the Product of Binomials
This is when we try to find the two binomial roots of a
polynomial.
See: https://gregmatmath1.blogspot.com/2020/05/week-5-day-1.html
Quadratic Equations
This is any equation of the form ax2 + bx + c =
0. You solve these by first factoring the equation, then solving each factor.
Alternative Strategies:
Backsolving
Try to plug in the answer choices to see if any work to satisfy the equation.
Picking Numbers
Try the FROZEN method to see if you can produce consistent
results.
Dealing with Word Problems
General approach:
1. Read through for gist.
2. Identify and label variables.
3. Translate English into mathematics.
4. Solve the equation(s).
5. Check your work:
a. check calculations
b. check that you answer the right question
c. check that you’ve made no typos/errors
Translation table (memorize)
This is just a matter of practice, and is not a problem
unique to non-native speakers. Don’t forget the “turnaround” words, and see the
expanded table on our memorization chart!
Treat each sentence as a separate equation the first time;
if that doesn’t work, then try combinations.
TAKE NOTES on what the question is asking for, so you can
remember to check that you answered the right question!
Dealing with Logic Problems
These are less about mathematical skill and more about
analytical and deductive logic. Try to arrange the information visually in a
picture, graph, diagram, or table. Organize and keep track of different
possibilities, and try to eliminate answers that do not make sense.
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