Mathematices

BASIC IDENTITIES IN ALGEBRA

Distributive Law

a(b + c) = ab + ac

Commutative Law

a + b = b + a

Associative Law

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

Difference of Squares

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

Sum of Cubes

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Difference of Cubes

a^3 - b^3 = (a - b)(a^2 + ab + b^2) 

POLYNOMIALS

A quadratic equation looks like this:

ax² + bx + c = 0  (where ‘a’ cannot be zero.)

Solving the equation means finding ‘x’ values that make the equation true. These ‘x’ values are called the roots  of the quadratic.

Quadratic equations can have 0, 1 or two roots.

The quadratic formula is derived from the general quadratic equation (below) by completing the square.

The general quadratic equation...

ax² + bx + c = 0

has roots...

           

Factoring

The binomial formulas.

Here are three algebraic formulas, the binomial formulas, which can be used for factoring:

You should check that these formulas work by multiplying out using the FOIL method.
Here is how to use these formulas for factoring purposes: Let's say we want to factor

We can write this polynomial as

and then notice that the terms match the second binomial formula for the values a=2x and b=3. Consequently,

and we have factored the polynomial completely. Note that x=3/2 is the only root, with multiplicity 2.
Here is another example: factor the polynomial

We can write the polynomial as the difference of two squares and then use the third binomial formula:

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The cube of a binomial.

There are similar formulas to factor some special cubic polynomials:
As an example, let us factor the polynomial

We can rewrite this polynomial as

Now it matches formula (5) with a=2x and b=3. Consequently

The polynomial has a triple root at x=3/2.

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Sums and differences of two cubes.

here are two more formulas to handle special cases of cubic polynomials:
Say, we like to factor . By formula (6), we can write

In this case the factorization is complete, since the polynomial is an irreducible quadratic polynomial.
What about the polynomial ? We first write this as the difference of two cubes, and then use formula (7):

Aside: Note that the factorization is still not complete. The Fundamental Theorem of Algebra tells us that it is possible to factor

further. Since you can see from the graph of this polynomial that it does not have real roots, the polynomial

can be factored into 2 irreducible quadratic polynomials. To find these two polynomials requires more familiarity with complex numbers; you can check that

Surface Area Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

 

Surface Area of a Cube = 6 a 2

(a is the length of the side of each edge of the cube)
In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a ² . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)
In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.
 

Surface Area of Any Prism

 (b is the shape of the ends)
Surface Area = Lateral area + Area of two ends
(Lateral area) = (perimeter of shape b) * L
Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)
 

Surface Area of a Sphere = 4 pi r 2

(r is radius of circle)
 

Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h

(h is the height of the cylinder, r is the radius of the top)
Surface Area = Areas of top and bottom +Area of the side
Surface Area = 2(Area of top) + (perimeter of top)* height
Surface Area = 2(pi r ²) + (2 pi r)* h
In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.
You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r²). Since there is both a top and a bottom, that gets multiplied by two.
The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.
Add those two parts together and you have the formula for the surface area of a cylinder.
Surface Area = 2(pi r ²) + (2 pi r)* h

 Volume Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a". "b³" means "b cubed", which is the same as "b" times "b" times "b".

Be careful!! Units count. Use the same units for all measurements. Examples

cube = a ³
rectangular prism = a b c
irregular prism = b h
cylinder = b h = pi r ² h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 pi r ² h
sphere = (4/3) pi r ³
ellipsoid = (4/3) pi r₁ r₂ r₃

Area Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

square = a ²
rectangle = ab
parallelogram = bh
trapezoid = h/2 (b₁ + b₂)
circle = pi r ²
ellipse = pi r₁ r₂
 

triangle =

one half times the base length times the height of the triangle

  

equilateral triangle =

 
triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C
triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)
regular polygon = (1/2) n sin(360°/n) S²
   when n = # of sides and S = length from center to a corner