Algebraic Formulas

This lists several important theorems (and explains some that are not as obvious), and a few important definitions.

*Pythagorean Theorem: a² + b² = c²

- This theorem is used to explain the lengths of sides in a right triangle. The two shorter sides are a and b, while the longer side, c, is called the hypotenuse. If one has any two values, it is possible to learn the value of any third side.

*Commutative property of addition: a + b = b + a

- This states that the order of any numbers being added may be interchanged, and it does not change the value of the equation.

*Associative property of addition: (a + b) + c = a + (b + c)

- This property states that placing parentheses in any part of an addition-only equation, which changes the order in which it is added, does not change the value of the equation.

*Identity property of addition: a + 0 = a

- This states that adding 0 to any number results in the same number.

*Inverse property of addition: a + (-a) = 0

- This states that if the opposite of any number is added to the original number, it equals zero. Ex. 1 + -1 = 0.

*Commutative property of multiplication: a × b = b × a

- This states that the order in which numbers are multiplied does not change their value.

*Associative property of multiplication: (a × b) × c = a × (b × c)

- This states that if parentheses are inserted anywhere into a multiplication-only equation, its value is unchanged.

*Identity property of multiplication: a × 1 = a

- This states that multiplying any number by one does not change its value.

*Inverse property of multiplication: a × 1/a = 1

- This property states that any number multiplied by its reciprocal (1 over that number) equals one.

*Distributive property: a(b + c) = ab + ac

- This states that if a number is multiplied by a quantity in parentheses, it is equal to that number times each of the numbers in parentheses.

*Reflexive property of equality: For any real number a, a = a

- This states that any number is equal to itself.

*Symmetric property of equality: For all real numbers a and b, if a = b, then b = a

- This states that if one number is equal to another number, then, in turn, the other number is equal to the first number.

*Transitive property of equality: For all real numbers a, b, and c, if a = b and b = c, then a = c

- This states that if one number is equal to another, and that number is equal to a third, then the first number must be equal to the third.

*Substitution property of equality: If a = b, then a may be replaced by b

- This states that since a and b are equal, they are interchangeable in any situation.

*Addition and subtraction properties of equality: If a = b, then a + c = b + c, and a - c = b - c

This states that if two numbers are equal, they will remain equal if the same number is added or subtracted from them.

*Multiplication and division properties of equality: If a = b, then a × b = b × c, and if c is not equal to zero, then a/c = b/c

- This states that if two numbers are equal, they will remain equal if they are multiplied or divided by the same number.

*Absolute value: If a > or = 0, then |a| = a; if a < 0,="" then="" |a|="-a">

- This states that if a is greater than or equal to 0, then the absolute value of a will be the same as its usual value. If a is less than zero, than a will be the opposite. Absolute values deal with knowing how much a number is, so it can never be negative. The absolute value of -5 is 5.

*Definition of slope: If two points, (x¹, y¹) and (x², y²) are given, the slope is equal to y² - y¹ / x² - x¹

- This states that if one is given two points, subtracting the second y coordinate from the first y coordinate, and dividing it by the second x coordinate minus the first x coordinate is equal to the slope.

*Definition of parallel lines: In a plane, nonvertical lines with the same slope are parallel.

*Definition of perpendicular lines: In a plane, two lines are perpendicular if and only if the product of their slopes is -1.

*Standard form of an equation: Ax + By = C

-A, B, and C are variables that may be substituted in. x and y are unknowns.

*Slope-intercept form of an equation: y = mx + b

- m = slope, and b = the y-intercept, which is the value that y equals when x is zero (substitute 0 into the equation for x, if the equation is in standard form, and solve for y to get the y-intercept).



*Point-slope form of an equation: y - y¹ = m(x - x¹), where (x¹, y¹) is a point on the line, and m is the slope.

*Value of a determinant: | a b | = ad - bc

| c d |

- This means that within a determinant (normally with one straight line on each side), when a is multiplied by d and subtracted by b multiplied by c, that is the value.

*Cramer's rule: The solution to the system ax + by = e is (x, y), where

cx + dy = f

| e b | | a e |

| f d | | c f |

x = ____ and y = ____

| a b | | a b |

| c d | | c d |

-This describes a system of equations, and a method of solving them. The variables can be anything.

*Addition of matrices:

| a b c | | j k l | | a+j b+k c+l|

| d e f | + | m n o| = | d+m e+n f+o |

| g h i | | p q r | |g+p h+q i+r |

-When two matrices are to be added, they first must have the same dimensions (in this case, three by three), then the corresponding parts must be added to come up with a new matrix with the same dimensions.

*Negative exponents

- a^-n = 1/a^n

- This is a symbol for exponents. It means that any number taken to a negative exponent is the same as one over the number with a positive exponent, and vice versa.

*Zero product property: For any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both.

- This is used in solving quadratic equations, when factoring is the method of solution.

*Sum and product of roots: If the roots of ax² + bx + c = 0, with a not equal to 0, are s1 and s2, then s1 + s2 = -b/a and s1 x s2 = c/a.

- This describes the nature of the roots if certain conditions are true, relating to a quadratic equation. S1 and s2 are the roots of the equation (found through factoring).

*Quadratic Formula: [-b ± (the square root of b² - 4ac)]/2a

- This is based on the quadratic equation mentioned in the previous problem. It is "x" that is solved for - one negative solution and one positive solution.

*Distance formula: When given two points, their distance on a plane is given as d = the square root of (x2 - x1)² + (y2 - y1)²

- Both "x2 and x1," as well as "y2 and y1" refer to the points taken from the line within the plane. "1 and 2" are simply a way to separate them.

*Midpoint formula: [(x2 + x1) ÷ 2], [(y2 + y1) ÷ 2]

*The remainder theorem: When a polynomial is divided, dividend = quotient x divisor + remainder.

*The factor theorem: The binomial x - a is a factor of the polynomial f(x) if and only if f(a) = 0.

*The location principle: Suppose y = f(x) represents a polynomial function and a and b are two numbers such that f(a) < 0="" and="" f(b)=""> 0. Then the function has at least one real zero between a and b.

- If a and b are greater than and less than zero, a zero of the function will fall between them if the equation is correctly factored.

*Fundamental theorem of algebra: Every polynomial equation with a degree greater than zero has at least one root in the set of complex numbers.

- If an equation exists, it must have at least one solution.

*Complex conjugates theorem: Suppose a and b are real numbers with b not equal to zero. If a + bi is a zero of a polynomial function, then a - bi is also a zero of the function.

- "˜I' represents an imaginary number (I is equal to radical negative one). This states that if a complex number is a solution of an equation, the opposite of it is too.

*Property of inverse functions: Suppose f and f^-1 are inverse functions. Then f(a) = b if and only if f^-1(b) = a.

*Inverse variation: y varies inversely as x if there is some nonzero constant k such that xy = k or y = k/x.

- This deals with the difference between x and y, and k, which is a variable which must be solved for by using x and y.

*Direct variation: y varies directly as x if there is some nonzero constant k such that y = kx. K is called the constant of variation.

*Definition of logarithmic: An equation of the form y = log b x, where b > 0 and b is not equal to one, is called a logarithmic function.

- b is subscript of the log. A logarithm is when b is taken to the x power.

*Product property of logarithms: For all positive numbers m, n, and b, where b is not equal to one, log b mn = log b n + log b m.

*Quotient property of logarithms: For all positive numbers m, n, and b, where b is not equal to one, log b m/n = log b m - log b n.

*Power property of logarithms: For any real number p and positive numbers m and b, where b is not equal to one, log b mP = p x log b m.

*Change of base formula: For all positive numbers a, b, and n, where a is not equal to one and b is not equal to one, log a n = (log b n)/(log b a).

*Definitions of trigonometric functions:

Sine = a/c Cosine = b/c Tangent = a/b

Cosecant = c/a Secant = c/b Cotangent = b/a

- A and B are the shorter legs of a right triangle (each side is labeled with the same letter as the opposite angle), and C is the hypotenuse.

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