Magoosh Algebra Notes

The Square of a Sum

(a+b)^2 (a+b squared) = a^2 +2ab +b^2

The Square of a Difference

(a-b)^s (a-b squared) = a^2 - 2ab - b^2

For MOST Quadratic Equations Follow these procedures

1) Get everything to one side of the equation and set everything to equal 0 2)Divide by GCF if necessary 3) Factor it into a product of linear binomials 4) Use the zero product property to create two separate linear equations and then solve for

Binomial

A binomial is an expression with exactly two terms.

A cubic term

A cubic term is a term with a cube of a single variable, y cubed or x cubed.

A linear term

A linear term, is a term with a single power of a variable, so there's no exponent written. So, just when you have a variable by itself or a number times a variable, that's a linear term. (The words linear and quadratic can describe individual terms, but they can also describe the entire expression, involving a single variable)

Monomial

A monomial is an expression with exactly one term. 5x^2 (5x squared); 43; X^2Y^3Z^5 (X squared, Y cubed and Z to the 5th)

Polynomial

A polynomial is an expression with any number of terms, involving only one variable. .

A quadratic term

A quadratic is a term with a squared variable. X squared or y squared. (The words linear and quadratic can describe individual terms, but they can also describe the entire expression, involving a single variable.)

Rational Expressions

A rational expression is a ratio, a fraction, of two algebraic expressions.

Term

A term is a product of constants and variables, including powers of variables. So for example, all of these are terms. --5, x, 6y^2 (six y squared), x^5y^6z^7 (x to the 5th, y to the 6th, z to the 7th) It could be just a single number by itself, a single variable by itself, or numbers times variables and that includes powers of variables. 1

Trinomial

A trinomial is an expression with exactly three terms.

Algebraic Expression

An expression is a collection of one or more terms joined by addition or subtraction.; don't have equal signs

Any EVEN power of x is a square of another power

Any EVEN power of x is a square of another power x^10 = x^5+5 = x^5x^5 (x to the fifth times x to the fifth) = (x^5)^2 (x to the 5th)squared

To find the Prime Factorization of a HUGE number or a TINY number we can write these numbers in terms of the difference of two squares and then simplify

Ex 1599: This is the differenct of 1600-1 = (40^2) - (1^2) (forty squared minus 1 squared --> (40 - 1)(40 + 1) ---> (39)(41) --->41*3*13 (These are the 3 prime factors of 1599. 2491 = 2500-9 = (50^2) - (3^2) = (50 - 3)(50 +3) = 47*53

The Distributive Law Multiplication over Addition:

Ex of Monomial times binomial P(Q+R) = PQ + PR or with binomials times binomial (p+q)(r+s) use FOIL

A linear expression

In a linear expression, the highest power of the variable is 1. So, 17x is a linear monomial. 3x minus 5 is a linear binomial. So one of them, it's a one term thing, the other is a two term thing. And notice we don't have many options here. A linear binomial must have one linear term and one constant term.

A quadratic expression

In a quadratic expression the highest power of the variable is 2. So 14x squared, that is a quadratic monomial. 2. X squared minus 4 is quadratic binomial with a quadratic term, x squared, and a constant term, negative 4. 2x squared plus 8x is a quadratic binomial with a quadratic term, 2x squared, and linear term, 8x. X squared minus 10x plus 25 is a quadratic trinomial with a quadratic term, the x squared, a linear term, negative 10x, and a constant term.

Like Terms

Like terms are any two terms with the same variable part. If they differ at all, they differ only in coefficients. The number one algebraic simplification rule, we can combine like terms by addition or subtraction.

Raising a power to a power

Multiply the two powers

Any equation with just variables x and y (with no exponents) has infinite number of points that when plotted out equal a line.... and written in a line of the x-y plane

No one can ask you to solve an equation with two variables b/c it's infinite

Solving a system of equations with substitution

Solve ONE of the equations for one of the variables This is best when at least one of the variables in one of the equations has a coefficient equal to 1 or -1

If we have a system of equations, with two unknowns, we generally can solve for unique values of x and y

Solve for these values by 1) substitution 2) elimination

The Absolute Value of a Number 'X' is the distance of that number from some other value

The absolute value of 'X' is The distance of 'X' from zero |X|; The absolute value of |x-5| is the distance of x from +5.

Coefficient

The coefficient is the constant factor of a term. It's '6' in 6y^2

The number 1 algebraic simplification Rule

The number one algebraic simplification rule, we can combine like terms by addition or subtraction.

System of equations = two equations each with two varialbes

The value of x & y must satisfy both equations, then the two equations represents the point at which the two lines intersect

Solvoing a system of equations with elimination

We can add two equations together and in some equations one variable would cancel... If they aren't already like this then we would multiply one equation by something and the other by something else so that the coefficients of the same variable is equal and opposite so that they cancel when we add them together.

Combining Inequalities

We can combine inequalities in the same direction --- a<b b<c then we can say a<b<c and we can say a<c We can add inequalities of the same dirction: ---- if a < b and c < d then (a+c) < (b + d) We can subtract inequalities in opposite direction ----- if a < b and d > c then (a-d) < (b-c) There are no rules regarding multiplying inequalities; Cannot do it and have any predictable results.

How to Simplify Rational Expressions

We must factor the expressions in the numerator and denominator and then cancel the common factors

Inequalities

When solving an inequality with a variable and 2 inequalities, we can add and subtract and dont' need to change the direction of the sign, when we divide or multiply (look at inequlities video I)

Variable

a letter that represents either a specific number or all numbers.

The Difference of two squares can always be factored

a^2 - b^2 (a squared minus b squared) = (a+b)(a-b)

Constant

constant is a number, or a symbol, such as pi, that doesn't change in value. So, just constant is just a fancy word we use for ordinary numbers.