Introduction to Factoring Trinomials - reverse of FOIL

YayMath.org  "Factoring means turn it into pieces you can multiply." 
►Accompanying worksheet for this video.



A trinomial has 3 terms.   x² + 2x - 63
A binomial has 2 terms.   x - 7
The 2 factors of a trinomial are binomials, and each can be written in parentheses.
(x + 9)(x - 7)  They are called binomial factors.
When these binomial factors are multiplied, they will equal the trinomial.

After checking for a greatest common factor and factoring it out, then you can factor the trinomial.

To easily determine signs when factoring trinomials:

1.  If the sign of the last term in a trinomial is negative, such as x² + 2x - 63
the signs between the terms in the binomial factors will be one positive and one negative.
Example 1:  
x² + 2x - 63
(x + 9)(x - 7)

Example 2:
x² - 2x - 63
(x - 9)(x + 7)

Do you see the difference?
• In the first example, only the last term in the trinomial was negative.
• In the second example, both the last term and the middle term were negative.
• So no matter what the middle term is, if the last term in a trinomial is negative, the signs between the terms in the binomial factors will be one positive and one negative.

So what does the sign of the middle term in the trinomial tell us?
It is what you use to determine which of the binomial factors will be positive, and which will be negative.
• In the first example, the middle term of the trinomial is positive (+2x), showing that the 9 in the binomial factors should be positive since it is greater than 7, because if you combine +9 and -7, you will get +2.
• In the second example, the middle term of the trinomial is negative (-2x), showing that the 9 in the binomial factors should be negative, because if you combine -9 and +7, you will get -2.


2.  If the sign of the last term of a trinomial is positive, the signs between the terms of the binomial factors will either be BOTH positive or BOTH negative.
• If the last term in the trinomial is + 63, the terms in the binomial factors will either be 
( __ - 7) and ( __ - 9), or they will be ( __ + 7 and ( __ + 9).
• Then the sign of the middle term of the trinomial will determine what they both will be.

Example 3:
x² + 2xy + y²
(x + y)(x + y)

Example 4:
x² - 2xy + y²
(x - y)(x - y)

If you use the FOIL method and multiply the Inner and Outer terms (from the binomial factors), you will get either both positive terms or both negative terms to add together, equaling the middle term of the trinomial.
In example 3, +xy and +xy will give you +2xy.
In example 4, -xy and -xy will give you -2xy.


►Found this video at Virtual Nerd: Determining Signs when Factoring a Trinomial

Multiplying Polynomials - FOIL and more

• Monomial - 1 term
• Binomial - 2 terms
• Trinomial - 3 terms
• Polynomial - more than 1 term

(1) YayMath.org - Foil method
►Accompanying worksheet for this video.
►Online quiz.


(2) Multiplying a binomial by a polynomial


(3) Multiplying vertically - I like this method!

Rules for Exponents

There are a lot of videos here!  About 35 minutes total, so I suggest taking more than one day to watch these, and taking notes as you do. The first several videos are shorter.

►Interactive lessons -- may be best to do after viewing the videos here.

Click Unit 5, Lesson 20, Play Lesson.  Click on the lesson titles on the left.

You may read along with the lesson in the left sidebar.

Note:  Lesson does not seem to work in Safari browser.

(1) Product Rule


(2) Quotient Rule


(3) Power Rule


(4) Using the rules


(5) Simplifying fractions with exponents


(6) Exponents, powers of base 10


(7) Level 1 Exponents, part 1 - the "why" of some of the above, and a few new concepts.


(8) Level 1 Exponents, part 2

In the last example, he leaves "3 to the negative 36th power." 3^-36, (since I can't write exponents on here except for °²³).  I have been taught and have seen taught in various videos online that you should try to not have a negative exponent in your answer.
In the 5th video, she showed how to get rid of negatives by moving the negative exponent to either the top or bottom of a fraction.  So the last answer could be written as a fraction: 1/3^36.  "1 over 3 to the 36th power."

Dividing Polynomials

YayMath.org
►Accompanying worksheet for this video.

Algebra 2 - Dividing Polynomials from Yay Math on Vimeo.

Systems of Equations: Inconsistent/Consistent; Independent/Dependent

When graphing systems of equations, you can figure out whether they are consistent or inconsistent.
If they are consistent, you then see whether they are independent or dependent.

(1) The differences


►If the lines are parallel to each other:
•there is no solution (There is no place of intersection - which would have been the solution, therefore no solution.)
•and they are inconsistent (have no points in common)
►If lines intersect:
•they have a solution such as (2, -4) or where ever they intersect, and that is what you write. "(2, -4)"
•they are consistent (have at least one point in common)
•and they are independent (of each other -- they go their own direction)
►If lines land in the same place, on top of one another:
•the solution is along the entire line - any and all points on the line will work as the solution, so you would say "entire line" or "infinite."  There is an infinite number of possible points along the entire line.
•they are consistent (have at least one point in common)
•they are dependent (do not go their own direction)

(2) Using substitution to tell the difference