Basic Algebraic Laws
Commutative Law For Addition:
a + b = b + a

ex. 5 + 7 = 7 + 5 =12


Commutative Law For Multiplication:


a x b = b x a

ex. 3 x 6 = 6 x 3 = 18


Associative Law For Addition:

(a + b) + c = a + (b + c)

ex. (4 + 6) + 5 = 4 + (6 + 5) = 15



Associative Law For Multiplication:

(a x b) x c = a x (b x c)

ex. (2 x 3) x 4 = 2 x
(3 x 4) = 24


Distributive Law:

a x (b +c) = (a x b) + (a x c)

ex. 3 x (2 + 5) =
(3 x 2) + (3 x 5)=21

Even though the above laws apply to addition and multiplication, they do not apply to subtraction and division.

For example:
7 - 5 = 2, which is not the same as 5 - 7 = -2.



Some Basic Algebra Rules



The additive inverse for a is (-a) for any constant.or variable. a + (-a) = 0
ex. 5 + (-5)= 0

The mulitplicative inverse for a is 1/a for any constant or variable. a x 1/a = 1
ex, 5 x 1/5 = 1

The identity element for addition is 0 for any constant or variable. a + 0 = a
ex, 5 + 0 = 5

The identity element for multiplication is 1 for any constant or variable. a x 1 = a
ex, 5 x 1 = 5



Rules for areas:


Rectangle or square = width x height

circle = 2 x pi x radius x radius

triangle = 1/2 x base x height



Polynomial Solutions for y = ax2 + bx + c


Sqrt is the same as square root here.


x =

-b +/- sqrt(b^2 - 4ac)
2a




Distance Formula between two points on a cartesian graph


Let the first point be (x1,y1) and the second point be (x2,y2). We use the Pythagorean thereom for a right triangle, using the length between the two x values as one side and the length between the y values as another side of a right triangle. The diagonal line, or shortest distance between the two points would be the hypotenuse of that right triangle. The distance, which we'll call z, between those two points would be:

z^2 = (x2 - x1)^2 + (y2 - y1)^2

We solve for z by taking the square root of both sides. Sqrt is the same as square root here.

We get z = sqrt[(x2 - x1)^2 + [(y2 - y1)^2 ]


Rules For Exponential Forms Under Multiplication And Division


xm times xn = x m + n ex. 23 x 24 = 23 + 4 = x7

xmdivided by xn = xm - n ex. 27 / 24 = 27 - 4 = 23


Rules For Exponentials Being Raised To A Power


(xm)n = xm^n = xmn ex. (x5)3 = x 5^3 = x15


Rules For Exponentials and 1/N Roots


(xm)1/n = x m/n ex. (x15)1/3 = x5


Steps For Solving For A Single Variable Equation


ex. 4x - 9 = 7 + 9x + 19


4x - 9 = 26 + 9x Combine Like Terms. In the case,
+9 = +9 add the 7 and the19. Add 9 to both sides to get rid of the 9 on the left
hand side. 9 is the additive
inverse of -9 and will cancel that term to zero.


4x + 0 = 35 + 9x
-9x = - 9x We need to get rid of the 9x
on the right side now, so we'll
subtract both sides by -9x.


-5x = 35

(-1)(-5x) = (-1)(20) Multiply both sides by -1 to get rid
of the minus sign in front of the -5x.


5x = -35


(1/5)(5x) = (1/5)(-35) Multiply both sides by 1/5, which
is the same as dividing both sides
by 5. We want to get rid of the 5 in
front of the x to solve for x.


x = -7 Our solution to the equation is x =
-7, If you plug -7 into the
original equation, the right side will
equal the left side of the equation.


How To Reduce Rational Expressions



x5 - 2x4 - 15x3

x6 + x5 + 12x4




(x3)(x2 - 2x - 15) Factor out the common factor
of x3 from the numerator and
(x4)(x + 7x + 12) factor out the common factor of
x4.



(x3)(x + 3)(x - 5) Factor out the remaining


polynomials in the numerator
(x4)(x + 4)(x + 3) and the denominator.


(x3)(x + 3)(x - 5) Cancel out any common
(x4)(x + 4)(x + 3) factors.


x - 5 We get the answer.

x(x + 4)