Maths
Binomial special products
"difference of squares" pattern: (a+b)(a-b)=a^2-b^2 (a+b)^2 =a^2 +2ab +b^2 (a−b)^2 =a^2 −2ab +b^2
Factoring quadratics: Perfect squares
(a+/-b)^2 = a^2 +/- 2ab + b^2 Expressions of this form are called perfect square trinomials Notice that both the first and last terms are perfect squares It is not necessary for the leading coefficient of a perfect square trinomial to be 1
isosceles
(of a triangle) having two sides of equal length.
Factoring quadratics in any form
- Factoring out the GCF (makes num easier) - Difference of Squares (If the first and last terms are perfect squares) - Perfect Square (If the first and last terms are perfect sqr and mid term is 2X their product) - Sum Product property - Grouping (two binomials even if you've to add +1)
factoring the polynomial.
-monomials by writing them as a product of other monomials. -GCF from polynomials using the distributive property -common binomial factors which resulted in an expression equal to the product of two binomials.
The number 1
1 does not fit into either category. It is neither prime nor composite.
Phi- golden ration
1.61
secant line
A line that passes through two points of a curve. The secant can be used to determine average rate of change between the two points it crosses the curve.
scientific notation
A number is written in scientific notation when there is a number greater than or equal to 1 but less than 10 multiplied by a power of 10.
Solving an addition equation using inverse operations
Addition and subtraction are inverse operations Inverse operations are opposite operations that undo or counteract each other.
equation
An equation is a statement that two expressions are equal equation with a variable, we call it an algebraic equation, our goal is usually to figure out what value of the variable will make a true equation.which is called "solution to the equation."
standard form
Ax+By = C where usually A,B, and C are integers *share a factor of one=cancel all common multiples Good for finding: y & x intercept In some cases (for example when solving systems of equations), we might want to bring an equation written in another form to standard form.
Associative property of addition:
Changing the grouping of addends does not change the sum. For example, (2 + 3) + 4 = 2 + (3 + 4)
Associative property of multiplication:
Changing the grouping of factors does not change the product. For example, (2 \times 3) \times 4 = 2 \times (3 \times 4)
Commutative property of addition:
Changing the order of addends does not change the sum. For example, 4 + 2 = 2 + 4
Commutative property of multiplication:
Changing the order of factors does not change the product. For example, 4×3=3×4
Composite numbers
Composite numbers have more than 2 factors. Composite numbers have more than one way that they can be divided into equal groups.
Factoring quadratics: Difference of squares
Every polynomial that is a difference of squares can be factored by applying the following formula a^2-b^2 = (a+b)(a-b) *perfect squares is a number that can be expressed as the product of two equal integersEX 9 is a perfect sqr b 9 = 3.3 The leading coefficient does not have to equal to 111 in order to use the difference of squares pattern. unles it is; then factor out a common factoctor
Factors
Factors are whole numbers that can evenly divide another number. Factors are whole numbers that can be divided evenly into another number without a remainder.
Factoring polynomials by taking a common factor(GCF)
Find the product of all their common prime factors Express each term as a product of the GCF and another factor. Use the distributive property to factor out the GCF. *the factored form is simply the product of that GCF and the sum of the terms in the original polynomial divided by the GCF. The common factor in a polynomial does not have to be a monomial.it can be a binomial
Factoring quadratics: leading coefficient = 1 second-degree trinomials factored from
Follows the sum-product pattern to find a & b just use solving equations with two variables method/ or the quadratic formula if x^2 is negative; factor out -1 However, not all trinomials with x^2 as a leading term can be factored
Greatest common factors
In integers: greatest integer that is a factor of both numbers. For example, the GCF of 12 and 18 is 6. We can find the GCF for any two numbers by examining their prime factorizations and circling the common ones: 12= 2.3.2 18=3.3.2 Notice that 12 & 18 have a factor of 2 and of 3 in common GCF= 3.2= 6 The variable part of the GCF for any two or more monomials will be equal to the variable part of the monomial with the lowest power of x
graph from slope-intercept
In order to graph a line, we need two points on that line. We already know that (0, b) is on the line Additionally, because the slope of the line is m, we know that the point (0+1, b+m) is also on the line
Writing decimals as fractions
Let x equal the decimal: {x = 0.7777... Set up a second equation such that the digits after the decimal point are identical: 10x = 7.7777... Subtract the two equations: 9x = 7 solve for x
Sequences {1, 3, 5, 7,...}
Ordered lists of numbers like these are called sequences. Each number in a sequence is called a term. Sequences usually have patterns that allow us to predict what the next term might be. ellipsis at the end show that the seq can be extended by using the pattern functions: We input a term number N, and the formula outputs the value of that term a(n); However, domain of sequences n—which is the set of all possible inputs of the function—is the positive integers. Notation:We prefer a(4)because it emphasizes that sequences are functions.
Prime numbers
Prime numbers are numbers with exactly 222 factors. A prime number's only factors are1 and the number itself. When there is only one possible way to divide a number into equal sized groups, that number is prime.
Slope
Slope is a measure of the steepness of a line.its inclination
equivalent systems.
Systems of equations that have the same solution produce an equivalent system by: replacing one equation by the sum of the two equations-graphing gets you two lines that are NOT parallel replacing an equation by a multiple of itself.-same line prove that they aren't equivalent by find a solution of one that is not a solution of the other.
Distributive property.or the distributive law of multiplication and division.
The math rule that allows us to break up multiplication problems
Identity property of multiplication:
The product of 111 and any number is that number. For example, 7 \times 1 = 7.
Square root
The square root of a number is the factor that we can multiply by itself to get that number. Finding the square root of a number is the opposite of squaring a number.
Identity property of addition:
The sum of 000 and any number is that number. For example, 0 + 4 = 4
General form of an absolute value equation f(x) = a [ x-h ] + k
The variable a tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables h and k tell us how far the graph/the vertex/the origin shifts horizontally and vertically.
intercepts
The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis.
forms of two-variable linear equation
There are three main forms of linear equations: slope-intercept;m is slope,b is y-intercept point-slope;m is slope, (a, b) is a point on the lie standard; A,B and C are constants
Writing fractions as decimals
To convert a fraction to a decimal, we divide the numerator by the denominator.
Monomial factorization
To factor a monomial means to express it as a product of two or more monomials. Completely factoring: To factor a coefficient is to factor an integer completely, we write it as a product of primes 30= 3.5.2 and expand the variable x^3 = x.x.x
intercepts from an equation
To find the y-intercept, let's substitute x by x= 0, into the equation and solve for y:
Factors and divisibility in integers
Two integers that multiply to obtain a number are considered factors of that number. 14=2⋅7 we know that 2 and 7 are factors of 14. One number is divisible: (a multiple of) by another number if the result of the division is an integer. 14/7=2 14 is divisible by 7 Their's a mutual relationship between factors and divisibility: 7 is a factor of 14 & 14 is divisibe by 7 *If a is a factor of b, then b is divisible by a, and vice versa.
Categories of numbers
We can divide almost all numbers into two categories: prime numbers and composite numbers.
negative exponents
We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:
intercepts from a table
We're given a table of values and told that the relationship between x and y is linear. The key is realizing that the x-intercept is the point where y=0,and the y-intercept is where x=0 To find the y-intercept, we need to "zoom in" on the table to find where y=0
linear equation
When solving an equation, our goal is to find the value of the variable that makes the equation true. here"s how we verify the solution: 3x+7 3⋅2+7put ?on topofevery = until last step where it's either = or /=
Factors and divisibility in polynomials
When two or more polynomials are multiplied, we call each of these polynomials factors of the product. One polynomial is divisible by another polynomial if the quotient is also a polynomial. EX. 2/x is not a polynomial
Standard Quadratic form
a 2nd degree polynomial ax^2+bx+c
The graph of a two-variable linear inequality
a line with one side shaded to indicate which x-y pairs are solutions to the inequality. *put it in slope-intercept form first /We shade below (Or above) because y is less or more than (or equal to) the other side of the inequality. /We draw a solid line (not dashed) because we're dealing with an "or equal to" inequality. The solid line indicates that points(coordintes) on the line are solutions to the inequality.
the elimination method
a technique for solving systems of linear equations. eliminating/canceling x or y terms by addition; multiplication(lcm) We can check our solution by plugging these values back into the the original equations=?.
The substitution method
a technique for solving systems of linear equations. we'll need to solve for either x of y in one of the equations. Now we can substitute the expression for x/y
polynomial
an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
monomial/polynomial
an expression that is the product of constants and nonnegative integer powers of x/ an expression that consists of a sum of monomials,
Surd
an irrational number, a number which cannot be expressed as a fraction or as a terminating or recurring decimal. It is left as a square root. It can also be a non-cube number left in cube root form and so on. . 1.414 is an approximation whereas Equation: sqrt{2} is exact. numbers left in square root form that are used when detailed accuracy is required in a calculation
Rational numbers
are numbers that can be expressed as a fraction of two integers.
Irrational numbers
are numbers that cannot be expressed as a fraction of two integers.
Whole numbers
are numbers that do not need to be represented with a fraction or decimal. Also, whole numbers cannot be negative. In other words, whole numbers are the counting numbers and zero.
Integers
are whole numbers and their opposites. Therefore, integers can be negative.
different equations call for different factorization methods.
bring quadratics to a factored form and solve them remove te common factor by dividing by it -square root -factor as: the product of two linear expressions (x+a)(x+b) using sum &/or product property -completing the square by adding to both sides or adding & subtracting same num &use 2a/2 to find a of perfet square factored as (x+a)^2+b then take sqr root
Simplifying a polynomial
combining like terms, that have same variables(letters & powers or constant terms)
the graphing method
either using : x & y intercepts by replacing both individually by 0 turning both equations to slope-intercept form-compare : slope & y-intercept(b) One solution. A system of linear equations has one solution when the graphs intersect at a point.-dif s & dif b No solution. A system of linear equations has no solution when the graphs are parallel.-same s dif b Infinite solutions. A system of linear equations has infinite solutions when the graphs are the exact same line.equations are equivalent and share the same graph-same s & same b
Factoring quadratics: leading coefficient /= 1
factoring b grouping-fisrt try to factor outa common fator to make it simpler with smaller coefficients a.b a+b split a & b of the sum group a & b apart with the first and last term factor out a monomial to have the same binomial *the quadratic formula may give you imaginary solutions OR/ Start by finding two numbers that multiply to ac and add up to b Use these numbers to split up the x-term. Use grouping to factor the quadratic expression. BUT/ it's not always possible to factor a quadratic expression of this form to (ax+b)(cx+d) using our method or any other where a,b,c,d are integers
Quadratic Formula
helps you solve quadratic equations ax^2+bx+c find the roots of a quadratic equation, i.e. the values of x where this equation is solved. when y = 0 the x-intercepts of the equation, i.e. where the curve crosses the x-axis. - If asked for the exact answer (as usually happens) and the square roots can't be easily simplified, keep the square roots in the answer, to find the number of soltions you only need to evaluate the discriminant b^2 - 4ac and it can be positive(two distinct real number solutions.), zero(repeated real number solution)., or negative(neither are real but imaginary)
The graph of a quadratic function
is a parabola, which is a "u"-shaped curve -vertex form : a( x-h )^2 + k (h, k) is the vertex x = h is the line/axis of symmetry a reveals whether the parabola concaves upwards or downwards (concavity = opening) to graph with this form we need another point -standard form: quadratic equation ax^2 + bx + c x-intercepts / roots by factoring; complt sqrt & taking sqrt; quadratic formula vertex: midpoint of the solutions then plug its x oordinate to find the y
Point-slope form
is a specific form of linear equations in two variables: y - b = m( x - a) from slope= y-b/x-a= m
slope-intercept form
is a specific form of two-variable linear equations. y = mx + b where m&b are real numbers The slope is m The y-coordinate of the y-intercept is b.. In other words, the line's y-intercept is at ( 0 , b )
A two-step equation
is an algebraic equation you can solve in two steps( usually including add/sub + mult/div. Once you've solved it, you've found the value of the variable that makes the equation true.
compound inequality
is an inequality that combines two simple inequalities. Let's take a look at some examples.
Why
m gives the slope : If we take two points where the change in x is exactly 1 unit, then the change in y will be equal to the slope itself. slope = change in y/1
Polynomial terminology
nth degree monomial, binomial, trinomial... polynomial rearranging according to term degree simplifying by adding like terms /terms coefficient non-neg integer exponent constant term
2 relatively prime numbers
numbers that only have 1 has their GCD/F or nothing in common from the prime factorization tree
Factoring two variable quadratics
rearrange the coefficient to make it in the form : ax^2+bx+c it can be with y as base as long as t is with 1 as leading coeffient
multiplying monomials
rearrange the factors because multiplication is commutative & add the exponents by polynomials- distributive property
Simplifying square roots
remove all perfect squares from inside the square root by factoring the number/variable
Inequalities
show the relation between two expressions that are not equal. <= Greater than or equal to
Proofs concerning irrational numbers
square roots of prime numbers are irrational there's an irrational number between any two rational numbers
Arithmetic sequences
the pattern involves adding or subtracting a number to each term to get the next term. the difference between consecutive terms is always the same. it's called "the common difference"
Geometric sequences
the ratio between consecutive terms is always the same. We call that ratio the common ratio.
Pie
the ratio of the circumference to the diameter of the circle: 3.14
In general
to determine whether one polynomial p is divisible by another polynomial q, or equivalently whether q is a factor of p, we can find and examine p(x)/q(x) If the simplified form is a polynomial, then p is divisible by q and q is a factor of p. *If 12x^2 is a fator of 6x then 6x is divisible by 12x^2(the denminator or divisor) so let's simplfy 6x/12x^2
More than just {a list of ordered numbers}
two new ways to represent arithmetic sequences: Recursive formulas The first term of a sequence The pattern rule to get any term in a sequence from the term that comes before it Explicit formulas. This formula allows us to simply plug in the number of the term we are interested in to get the value of that term; it can have multiple equivalent formulas; BUT only the standard form gives us the first term and the common difference. Formulas give us instructions on how to find any term of a sequence.how to find a(n) for any possible n formulas use n to represent any term number and a(n) to represent the nth term of the sequence
linear/quadratic equations
variable raised to 1st power/2nd power Quaratics : this equation has two solutions; positive and negative square root which is the inverse operation of squaring Unlike linear equations, you don't expand it when it's factored out so that taking the square root is easier This symbol means "plus or minus," and it is important because it ensures we catch both solutions
