Semester Project

Factoring Methods

Special Forms

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Some Special forms are listed below. when factoring the goal is to recognize a solving form and use the equality to simplify the polynomial. There are geometric properties that explain why, and how these special forms work. Below there is material that shows the connections to these geometric properties.

Perfect-Square Trinomials

a^{2} + 2ab + b^{2} ={(a+b)}^{2}

a^{2} - 2ab + b^{2} ={(a-b)}^{2}

Example :

(I couldn't find any material that can connect this to a geometric proof, so I will walk through the process with an example.) To demonstrate this special form of factoring I will use the equation

x^{2} + 10x + 25

. When we are trying to identify whether an equation is a perfect square trinomial we can work one term at a time to figure out if this type of factoring technique will work.

x^{2}

is the square of x, and 25 is 5 squared. Multiplying these gives me 5x. To get the middle term, 10x, I multiply 5x by 2. In order to make a perfect square trinomial we must match the term 2 to 5x like we just did. This means our equation can turn into

{(x)}^{2} - 2(5x) + {(5)}^{2}

. When finding the end result, whatever the starting binomial's square root is will be the first term in the end equation. The second term will be the square root from the third binomial. This means that the answer is

{(x+5)}^{2}

(We can check our answer by multiplying our answer out.) (Perfect-Square Trinomials, 2021)

Diffrence of Squares:

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

This applet was not made by me. This is Henri Picciotto's. link to Henri Picciotto's page.

What is the area of the two squares?
What is the area of the rectangle that is formed after using the slider?

Difference of cubes:

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

Sum of cubes:

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