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Here are a few recommended readings before getting started with this lesson.
Factoring
Tearrik is watching the clock on the wall just waiting for the school bell to ring so he can prepare for a fun weekend with friends and family. Just before the bell rang, his math teacher assigned the following challenge.
At first glance, he thought that the task was very simple since there is only one number whose cube is $125.$ Is Tearrik right? Are there no more solutions to the equation? Find all the solutions to the equation.
In Tearrik's courses, he previously learned that some reallife situations such as saving a constant amount of money weekly or shooting a basketball can be modeled by linear and quadratic equations, respectively.
Now, he wonders if there are situations involving other types of equations. Specifically, Tearrik wants to know if an equation could contain a polynomial. Luckily for Tearrik, his teacher is planning on introducing the topic next class.
On the weekend, Tearrik and his friends decided to go to the amusement park to have some fun.
$LHS−7=RHS−7$
Rearrange equation
$LHS⋅5=RHS⋅5$
Zero Property of Multiplication
Distribute $5$
Calculate quotient
Substitute values
Calculate power and product
$(a)=a$
Subtract term
Calculate root
Factor out $2$
$ba =b/2a/2 $
State solutions
$(I), (II):$ Add and subtract terms
$LHS⋅5=RHS⋅5$
Multiply
Distribute $5$
Calculate quotient
$LHS−44=RHS−44$
Rearrange equation
$LHS⋅(1)=RHS⋅(1)$
Zero Property of Multiplication
Distribute $(1)$
Associative Property of Addition
Factor out $t_{2}$
Factor out $(t−6)$
After arriving home and feeling excited about solving some polynomial equations at the amusement park, Tearrik sees a note written by his sister. Tearrik gets right to his homework so he can finish in time to watch a movie with his family!
Tearrik's homework asks him to factor a pair of polynomial equations.
$48x_{9}=12x_{7}⋅4x_{2}$, $108x_{7}=12x_{7}⋅9$
Factor out $12x_{7}$
Substitute expressions
Factor out $3x_{5}$
While checking the factorization methods he knows so far, Tearrik notices that he has a formula for factoring the sum and difference of two squares.
Sum of Squares  Difference of Squares 

$a_{2}+b_{2}=(a+bi)(a−bi)$  $a_{2}−b_{2}=(a+b)(a−b)$ 
However, Tearrik wonders if a similar formula exists for the sum of two cubes. The good news is that such a formula does exist and, along with the Zero Product Property, is useful for solving polynomial equations.
The sum of two cubes can be factored as the product of a binomial by a trinomial.
$a_{3}+b_{3}=(a+b)(a_{2}−ab+b_{2})$
Notice that the binomial is the sum of the bases $a$ and $b,$ and the trinomial is the sum of the bases squared minus the product of the bases.
Multiply parentheses
$a_{m}⋅a_{n}=a_{m+n}$
Commutative Property of Addition
Associative Property of Addition
Add and subtract terms
After reading the formula and having in mind that a cube can also be thought of as a threedimensional object, Tearrik wondered whether there is a geometric way of deducting the formula. Indeed, there is one way of visualizing the formula geometrically. First, consider two cubes, one of side $a$ and another of side $b.$
Next, place the small cube on top of the bigger one and complete the missing parts to form a prism.In need of a break from such a fun weekend, Tearrik decided to just relax in his room. Looking around, he sees a die and a Rubik's cube that have been laying around forever. He wonders about the sum of their volumes. He knows that each side of the die measures $2$ centimeters but does not know the dimensions of the Rubik's cube.
Substitute values
Calculate power and product
Subtract term
Write as a power
$a_{m}⋅b_{m}=(a⋅b)_{m}$
$a_{3}+b_{3}=(a+b)(a_{2}−ab+b_{2})$
Calculate power and product
Zero Property of Multiplication
$(I):$ $LHS−7=RHS−7$
$(I):$$LHS/2=RHS/2$
Use the Quadratic Formula: $a=4,b=14,c=49$
Calculate power and product
$(a)=a$
Subtract term
$a =a ⋅i$
Split into factors
$a⋅b =a ⋅b $
Calculate root
Factor out $2$
$ba =b/2a/2 $
Write as a sum of fractions
Equation  Solutions 

$8x_{3}+350=7$  $x_{1}x_{2}x_{3} =27 =47 +473 i=47 −473 i $

Substitute values