Chapter 3:

Polynomial Equations

As it turns out, scaling by a negative factor, isnʼt a “negative scaling” at all. Itʼs a combination of a vertical reflection about the x-axis, followed by a vertical scaling of a positive factor.

The 4 step numbers in the example below, are also labels for the 4 graphs on the right. 1)

-2x + 3 = 9

original polynomial equation

! 2) !

after translating down 3 (subtracting 3 from both sides)

2x = -6

after reflecting about the x-axis

! 3)

!

! 4)

!

-2x = 6

(multiplying both sides by -1) x = -3

!

after scaling vertically by 1/2 (dividing both sides by 2)

4

3

2

1

Since all of the transformations are vertical, the missing horizontal x value for each solution, always stays the same. This is why algebraically, all of the above steps are equivalent equations, and why we, “solve for x.”

20

Chapter 3:

Polynomial Equations

Knowing multiplications are scalings, and negatives are reflections, we can explain multiplying with negatives. 3 • 2 = 6 , is just the point 3 on a number line, being scaled by 2 (made twice as far from the origin).

-3 • 2 = -6 , is just the reflection of 3 on a number line, being scaled by 2 (made twice as far from the origin).

3 • -2 = -6 , is just the point 3, being reflected and scaled by 2 (turned into a -3 and made twice as far from the origin).

-3 • -2 = 6 , is just the reflection of point 3, being reflected and scaled by 2 (turned into a 3 and put twice as far away).

$ To explore more examples, run the applet “multiply.html” .

21

Chapter 3:

Polynomial Equations

22

Knowing additions/subtractions are translations, and negatives are reflections, we can explain addition/subtraction with negatives. 5 + 2 = 7 ( and 5 - -2 = 7 ) , is just a 2 unit translation to the right from point 5 (translate left 2, then reflect about 5).

-5 + 2 = -3 ( and -5 - -2 = -3 ) , is just a 2 unit translation to the right from point -5 (translate left 2, then reflect about -5).

5 - 2 = 3 ( and 5 + -2 = 3 ) , is just a 2 unit translation to the left from point 5 (translate right 2, then reflect about 5).

-5 - 2 = -7 ( and -5 + -2 = -7 ) , is just a 2 unit translation to the left from point -5 (translate right 2, then reflect about -5).

$ To explore more examples, run the applet “add.html” and/or “sub.html” .

Chapter 3:

Polynomial Equations

23

Remembering that additions are vertical translations, negatives are reflections about the x-axis, and multiplications are vertical scalings, gives a clearer picture of solving degree 1 polynomial equations. Below is a snapshot of a program, which will randomly generate 10 degree 1 polynomial equations for you to solve.

! # "

Run program 303 , until you can successfully solve at least 8 of the 10 degree 1 polynomial equations.

Use a copy of the pdf “programnotes.pdf” , to record the original equations, and their 2 (or 3) equivalent forms.

After solving at least 8 equations, and recording them along with their 2 (or 3) equivalent forms, go to the next page.

Chapter 3

The 4 step numbers in the example below, are also labels ... 3 • 2 = 6 , is just the point 3 on a number line, being scaled by 2 (made twice as far from the origin).

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