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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 real-life 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)
\AssociativePropAdd
Factor out t2
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.
48x9=12x7⋅4x2, 108x7=12x7⋅9
Factor out 12x7
Substitute expressions
Factor out 3x5
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
a3+b3=(a+b)(a2−ab+b2)
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 |
---|---|
8x3+350=7 | x1x2x3=-27=47+473i=47−473i
|
Substitute values
\CommutativePropAdd
Subtract term
ba=b⋅2a⋅2
Add fractions
Add and subtract terms
a0=0
Substitute expressions
Factor out (a−b)
\CommutativePropAdd
The difference of two cubes can be factored as the product of a binomial by a trinomial.
a3−b3=(a−b)(a2+ab+b2)
Notice that the binomial is the difference of the bases a and b, and the trinomial is the sum of the bases squared plus the product of the bases.
Multiply parentheses
am⋅an=am+n
\CommutativePropAdd
\AssociativePropAdd
Subtract terms
Rewrite 64 as 43
a3−b3=(a−b)(a2+ab+b2)
Calculate power
Use the Quadratic Formula: a=1,b=4,c=16
Calculate power and product
Subtract term
-a=a⋅i
Split into factors
a⋅b=a⋅b
Calculate root
Factor out 2
ba=b/2a/2
Equation | Solutions |
---|---|
x3−27=37 | x1=4 x2=-2−23i x3=-2+23i |
Real Solutions | Imaginary Solutions |
---|---|
x=4 | x=-2−23i |
x=-2+23i |
Tearrik and tío Angelito are now taking a break from woodwork. Tío Angelito tells Tearrik to pull up a chair — he has a story to tell. Way back in the day, he used to go diving off the coast. Being that he loves math, when he dives he tries to follow the trajectory of a polynomial.
By following the given polynomial, he was able to model the trajectory of one of his favorite dives. In this polynomial, D(t) represents the depth, in meters, at which tío Angelito was t minutes after he started diving. Negative values of D(t) mean that he was underwater.
D(t)=-90
LHS+90=RHS+90
Rearrange equation
t | 1.5 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 11.2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
f(t) | ≈77 | 0 | -40 | 0 | 54 | 80 | 60 | 0 | -70 | -96 | 0 | ≈42 |
Given those four intercepts, it can be determined that there were four moments in which tío Angelito was exactly 90 meters underwater — namely, at 2, 4, 8, and 11 minutes after he began his dive.
LHS+200=RHS+200
Rearrange equation
t | 1.5 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 11.2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
g(t) | ≈187 | 110 | 70 | 110 | 164 | 190 | 170 | 110 | 40 | 14 | 110 | ≈152 |
LHS+150=RHS+150
Rearrange equation
t | 1.5 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 11.2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
h(t) | ≈137 | 60 | 20 | 60 | 114 | 140 | 120 | 60 | -10 | -36 | 60 | ≈102 |
Each of the questions can also be solved by analyzing the graph of y=D(t). As done in the previous parts, begin the process of graphing the equation by making a table of values.
t | 1.5 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 11.2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
D(t) | ≈-13 | -90 | -130 | -90 | -36 | -10 | -30 | -90 | -160 | -186 | -90 | ≈-48 |
Based on the drawn graph, some conclusions can be made.
x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
y | ≈658 | ≈507 | ≈333 | ≈183 | ≈78 | ≈21 | ≈1 | ≈0.5 | ≈0 | ≈-15 | ≈-47.5 | ≈-85 |
According to the table, there is only one sign change, which means that between -4 and 7 there is only one x-intercept — at x=4. Therefore, the graph of the polynomial should look as follows.
Furthermore, from -3 to the left the graph goes up, so it is expected to continue going up to the left of x=-4 as well. Similarly, from 6 onwards the graph decreases, so it is expected to continue descending to the right of x=7.
This is all that can be deduced from the table of values. However, some things are mistaken. According to a graphing calculator, the graph of the polynomial is the following.
As seen, the graph intersects the x-axis twice between 2 and 3. Also, the behavior on both ends is contrary to the one previously found. Therefore, the table of values should include more values and also consider decimal numbers.
A table of values might not precisely reflect all the characteristics of a graph.