Intro to Algebra
For our purposes, algebra is the part of math that involves variables
Some terms:
Variable - a letter that represents either a specific number or all numbers
The point is to find or use patterns true for all numbers
In algebraic equations, the variable represents one or two initially unknown values, and the point is to solve for those specific values.
Constant - a number, or a symbol, such as pi, that does not change in value
Term - a product of constants and variables, including powers of variables e.g. 5,x,6y^2
Coefficient - the constant factor of a term when no constant is written, the coefficient is 1
Expression - a collection of one or more terms joined by addition or subtraction NOTICE: expressions do not have equal signs
Monomial单项式 - an expression with exactly one term
5x^2
Binomial - an expression with exactly two terms
X+5
Trinomial - an expression with exactly three terms
X^2+4x-21
Polynomial - an expression with any number of terms, involving only one variable
Linear - a term with a single power of a variable
Quadratic - a term with the square of a single variable
Cubic - a term with the cube of a single variable
This is tricky. The words linear and quadratic can describe individual terms, but they can also describe entire expressions involving a single variable.
In a linear expression, the highest power of the variable is 1
In a quadratic expression, the highest power of the variable is 2
Simplifying Expressions
Multiplying Expressions
Multiplication is commutative & associative 3xy = 3yx
Distributive Law a(b+c) = ab+ac
FOIL Method
First+Outer+Inner+Last = product of the binomials
Square of a Sum: (a+b)^2 = a^2+2ab+b^2
Square of a Difference: (a+b)^2 = a^2-2ab+b^2
Factoring -GCF
P(Q+R) = PQ + PR
Left-> Right : distributing
Right -> Left : factoring out
Factoring - Difference of Two Squares
a^2-b^2 = (a+b)(a-b)
The material covered by this pattern expands considerably when we realize this: any even power of x is the square of another power
Note that there is no way to factor the sum of two squares
Factoring - Quadratics
The linear coefficient of the quadratic is SUM of these two unknowns, and the constant term of the quadratic is the PRODUCT of the two unknowns
Notice that this method only works if the coefficient of x^2 is 1. If the quadratic coefficient is something other than 1, the chances are very good that one of the other factoring methods can be used (Difference of Two Squares, GCF, etc.)
Factoring - Combined
Advanced Numerical Factoring
Remember that the Difference of Two Squares pattern can be powerful in finding the prime factorization of large numbers or in simplifying decimals just less than one.
Factoring - Rational Expressions
A rational expression is a ratio, a fraction, of two algebraic expressions
1) we learned how to simplify rational expressions by factoring the numerator & denominator and canceling the common factor
2) we also learned how to simplify by separating the numerator into two parts
Basic Equation Solving
1) in solving eq., our goal is to find the unknown value of the variable
2) mathematically, it is always legal to do any arithmetic operation to both sides of an eq.
3) strategically, we undo the order of operations to isolate the variable
4) if the variable appears on both sides, we begin by collecting all terms with the variable on one side
Eliminating Fractions
1) we can ‘undo’ multiplication by a fraction by multiplying by the fraction’s reciprocal
2) when we have multiple fractions added or subtracted, we can clear the fractions by multiplying by the LCM of the denominators
3) we can simplify a complex fraction by multiplying the numerator&denominator of the big fraction by the LCM of all the denominators of the little fractions.
Quadratic Equations
To solve most quadratic equations on the test:
1) gather all terms on one side, set equal to zero
2) divide by any numerical GCF
3) factor
4) use the Zero Product Property to separate into two linear equations, and solve
Zero Product Property:
If A*B = 0, then A = 0 or B = 0
Quadratics usually have two solutions, but can have one solution or no solution
Two Equations, Two Unknowns - 1
1) a system of equations, two equations with two var., typically has a single unique solution
2) we can solve with either substitution or elimination
3) substitution works best when one of the variables has a coefficient of+/-1
Two Equations, Two Unknowns - 2
1) elimination is more convenient if the two coefficients of the same variable are equal or opposites, or if one is a multiple of the other
2) if you are asked to find the value of an expression, you will almost always be able to find that without finding the values of the individual variables
System - Number of Solutions
1) a system of two equations with two unknowns may have one solution, no solution, or infinitely many solutions
2) if you solve and get an ALWAYS TRUE equation (e.g. 12=12), then the system has infinitely many solutions
3) if you solve and get a NEVER TRUE equation (e.g. -2=9), then the system has no solutions
Three Equations with Three Unknowns
Strategy
1) pick two of the three equations and, using either substitution or elimination, eliminate one variable.
2) pick another pair of the original equations and, similarly, eliminate the same variable
3)now, use two-equations-with-two-unknown techniques
4) plug in to any original equation to find the value of the third variable.
Absolute Value Equations
1) we solve the equation |A| = B by splitting it into two equations: A = B OR A = -B
2) remember: we have to isolate the absolute value before we can split it into the two ‘or’ equations
3) once we solve for answers we have to check that each works in the original equations; in this way, we eliminate extraneous外部的 roots.（注意注意）
Function Notation
1) the notation f(x) = expression means x is the ‘input’. We plug the value of x into the expression, and the resultant value of the expression is the ‘output’
2)we can also plug expressions into functions as the ‘input’. In the formula for the function, we replace every x with the expression that we can plugging in
Strange Operations
1) the test will give you strange operators that you and everyone else has not seen before. Do not panic.
2) the test has to give the rule for the operator. Simply follow the rule.
3) remember to find the numerical value of expressions inside parentheses first. You can replace the parentheses with the overall value of that expression.
Inequalities - 1
1) we can add & subtract with inequality, exactly as we do with equations
2) we can multiply and divide inequalities by positive numbers
3) if we multiply or divide an inequality by a negative number, this reverses the direction of the inequality
Inequalities - 2
1) we can combine inequalities (a<b, b<c) in the same direction (a<b<c)
2) we can add inequalities in the same direction: if a<b and c<d, then (a+c)<(b+d)
3) we can subtract inequalities in opposite directions: if a<b and d>c, then (a-d)<(b-c)
4) there is no general rule for the multiplication or division of inequalities
5) (any positive) > (any negative)
6) adding a positive makes number greater; subtracting a positive make it less
Absolute Value Inequalities
The distance of x from +p = |x - p|
The distance of x from -p = |x + p|
Simplifying with Substitutions
1) if one expression is repeated in an equation, we can choose a single variable for the expression, solve for its value, and then use that to solve for the original variable
2) if an algebraic expression has multiple parts, such as a compound fraction, we can solve for numerical values part by part
