Algebra Honors ( period 6)

Solving Inequalities  ( All 4 Op's) 5-1 &5-2

Solving Inequalities 5-2

Again you will use your equation skills but this time use the Multiplicative Inverse Property as you would if you were balancing an equation.

ONE MAJOR DIFFERENCE FROM EQUATIONS:

When you multiply or divide by a NEGATIVE coefficient (to balance) you must SWITCH the inequality SYMBOL (this does NOT apply to adding or subtracting negatives) You must rewrite the problem as you divide by a negative as shown below:

If you want to understand why:    3 < 10      you know that is true

Now multiply both sides by -1 ( multiplication property of equality lets you do that)
but you get -3 < -10 but THAT IS NOT TRUE

You have to SWITCH THE SYMBOL to make it true  -3 > -10

REMEMBER: When you MULTIPLY or DIVIDE by a NEGATIVE, the symbol SWITCHES!

Doing 2 Steps with Inequality Signs:  same as equations except make sure you switch the symbols if you multiply or divide by a negative. (Rewrite that portion of the problem—as you multiply or divide)  Always finish with the variable on the left!

Check with whatever solution is the easiest in the solution set. If 0 fits—use it! 

NEVER USE THE BOUNDARY NUMBER   for instance if your solution was x ≤  4  You could check with any number less than 4—BUT NEVER USE 4!

With two steps—before you start, you may want to clear fractions or decimals but if you don’t mind using them—just get started with the checklist below: If you want to clear them- you should do that right after you distribute ( which is between Steps 1 and 2 below)

1     1, Do Distributive Property first (if necessary) do it carefully

2      2.  Combine like terms on each side of the “WALL”

3   3.  “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term

4     4.   Add or subtract

5      5.  Multiple or divide ( only FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance)

     6.   Make sure the variable is on the LEFT side when finished.

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.

Algebra ( Periods 1 & 4)

Solving Inequalities ( All 4 Op's) 5-1& 5-2

Solving Inequalities 5-2

Again you will use your equation skills but this time use the Multiplicative Inverse Property as you would if you were balancing an equation.

ONE MAJOR DIFFERENCE FROM EQUATIONS:

When you multiply or divide by a NEGATIVE coefficient (to balance) you must SWITCH the inequality SYMBOL (this does NOT apply to adding or subtracting negatives) You must rewrite the problem as you divide by a negative as shown below:

If you want to understand why:    3 < 10      you know that is true

Now multiply both sides by -1 ( multiplication property of equality lets you do that)
but you get -3 < -10 but THAT IS NOT TRUE

You have to SWITCH THE SYMBOL to make it true  -3 > -10

REMEMBER: When you MULTIPLY or DIVIDE by a NEGATIVE, the symbol SWITCHES!

Doing 2 Steps with Inequality Signs:  same as equations except make sure you switch the symbols if you multiply or divide by a negative. (Rewrite that portion of the problem—as you multiply or divide)  Always finish with the variable on the left!

Check with whatever solution is the easiest in the solution set. If 0 fits—use it! 

NEVER USE THE BOUNDARY NUMBER   for instance if your solution was x ≤  4  You could check with any number less than 4—BUT NEVER USE 4!

With two steps—before you start, you may want to clear fractions or decimals but if you don’t mind using them—just get started with the checklist below: If you want to clear them- you should do that right after you distribute ( which is between Steps 1 and 2 below)

1     1, Do Distributive Property first (if necessary) do it carefully

2      2.  Combine like terms on each side of the “WALL”

3   3.  “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term

4     4.   Add or subtract

5      5.  Multiple or divide ( only FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance)

     6.   Make sure the variable is on the LEFT side when finished.

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.

Math 6A ( Periods 2 & 5)

Absolute Value 6.4

The absolute value of a number is the distance between the number and ZERO on a number line. The absolute value of a number a is written as │a│.

│-2│ = 2       │2│  = 2

  │a│ is read “ the absolute value of a.”

Distance is always positive OR zero!

When you write the notation for the absolute value, it means “take the absolute value of the number inside the symbols! “

When graphing  │-5│  first find the value of │-5│, which is 5, then graph it.  Make sure to identify the graph with the given number. In this case, │-5│

So we could conclude that  │-5│> 2.

Write these numbers in order from least to greatest.

│-12│, -8, -10, │6│, │-4│

We would get:

-10, -8, │-4│, │6│, │-12│

Which is greater  -50 or 25?   25

Now, which of those two has the greater absolute value?

 This time its -50 since

│25│= 25   and │-50│=50

The coldest possible temperature is called absolute zero. It is represented by 0 K on the Kelvin temperature scale.  See Page 273

Tell whether the statement is always, sometimes, or never true.

The absolute value of a number is greater than the number.  Sometimes- If the number is negative then its absolute value is greater, but if it is positive or zero then it is equal to its absolute value.

The absolute value of a negative number is positive. Always- The absolute value is the positive distance from zero on a number line.

The absolute value of a positive number is its opposite. Never- The absolute value of appositive number is the number itself

Math 6A ( Periods 2 & 5)

Fractions and Decimals on the Number line 6.3

How can we use a number line to compare positive and negative fractions and decimals?

Can you name a number between 1 3/6 and 1 4/6? Did you use equivalent fractions?

You can graph negative fractions and decimals in a similar manner to what we did with integers!  

Compare -1/2 and -3/4 
Notice -3/4 is farther to the left from 0. -1/2 is to the right of -3/4

So -1/2 > -3/4

Compare -4 5/6 and – 4 1/6

When you graph these two numbers, you notice that -4 5/6 is to the left of -4 1/6

So -4 5/6 < -4 1/6

Similarly with decimals

Comparing -3.08 and -3.8

-3.08 is to the right of -3.8 so

-3.08 > -3.8

Math 6A( Periods 2 & 5)

Comparing & Ordering Integers 6.2

On a  horizontal number line, numbers to the left are less than numbers to the right. Numbers to the right are greater than numbers to the left.

On a vertical number line, numbers below are less than numbers above. Numbers above are greater than numbers below.

When ordering numbers from least to greatest graph each number on a number line , then it is easy to list them! Write the integers as they appear on the number line from left to right!

Math 6A ( Periods 2 & 5)

Integers 6.1

Positive Numbers  are greater than 0. They can be written with or without a positive sign (+)

Negative Numbers are less than 0. They are written with a negative sign (-).

Two numbers that are the same distance from 0 on a number line, but on opposite sides of ZERO are called opposites. The opposite of 0 is 0.

Integers are the set of whole numbers and their opposites.

Note: ZERO (0) is neither positive NOR negative

Math 6A ( Periods 2 % 5)

Adding and Subtracting Decimals  2-4

Decimals may be added or subtracted using the same rules as whole numbers

Write the given numbers one above the other with the decimal points in line.

Annex zeros to get the same number of decimal places and then add or subtract as if the numbers were whole numbers.

Place a decimal point in the number for the sum or difference in position under the decimal points in the given numbers.

Add 6.47 + 340.8 + 73.523

STEP 1	STEP 2	STEP 3
6.47	6.470	6.470
340.8	340.800	340.800
+   73.523	+  73.523	+  73.523
	420 793	420.793

The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. Use estimates as a habit to check if your answer is reasonable.

To find an estimate, first round the highest place value of the smallest number, then compare.

Add 8.574 + 81.03 + 59.432. Then estimate to check your answer.  What is the highest place value of the smallest number?          9 + 81 + 59  = 149

Math 6A ( Periods 2 & 5)

Dividing Mixed Numbers 2.3

Dividing a Mixed Number by a Fraction

Write each mixed number as an improper fraction. Then divide as you would with a proper fraction.      Find   

Write   as the improper fraction            so it becomes          

but that becomes    because we multiply by the reciprocal of 3/8 which is 8/3  Simplify first—if you can                = 6

Dividing Mixed Numbers

Find                          ALWAYS ESTIMATE FIRST   4÷ 2 = 2

Write each mixed number as an improper fraction

  but that becomes      or   notice you can simplify so that before you multiply you have       =  

Real Life Application

One serving of tortilla soup is 1 2/3 cups. A restaurant cook makes 50 cups of soup. Is there enough to serve 35 people? Explain your thinking.

Divide 50 by 1 2/3  to find the number of available servings.

      If we simplify first 

There will not be enough to feed 35 people. There is only enough to feed 30 people.

Math 6A ( Periods 2 & 5)

Dividing Fractions 2.2

Two numbers whose product is 1 are reciprocal. 

To write the reciprocal of a number, write the number first as a fraction—then invert the fraction.

so the reciprocal of a fraction        

Dividing Fractions

To divide a number by a fraction, multiply the number by the reciprocal of the fraction.

Examples:
Numbers      

Algebra   

Find                                           

Dividing a Fraction by a Whole Number

Find              Remember:  always invert the 2^nd number – the divisor

so this becomes    

Algebra/Algebra Honors: Properties of Numbers 1-3

CHAPTER 1-3: Properties (2 days)

Re-introducing you to lots of old friends today!

WHAT ARE PROPERTIES?  They are characteristics of math operations that can be identified

WHY ARE THEY YOUR FRIENDS? (BFFs or Best Friends Forever) You can count on properties.

They always work. There are NO COUNTER EXAMPLES!

THEY ALLOW YOU TO WRITE EQUIVALENT EXPRESSIONS FOR AN EXPRESSION AND THE NEW EXPRESSION MAY BE EASIER TO USE!!

COUNTEREXAMPLE = an example that shows that something does NOT WORK
(counters what you have said)

PROPERTIES ARE THE EXCEPTIONS TO AUNT SALLY

Some properties give you a choice when it's all multiplication OR all addition
There are no counterexamples for these two operations.
BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples!  10 - 2 does not equal 2 - 10
15 ÷ 5 does not equal 5 ÷ 15)

JUSTIFYING

Because you can ALWAYS count on PROPERTIES, you can use them to JUSTIFY what you do mathematically.
JUSTIFY = giving a reason for doing what you did, and those reasons are your BFFS, the properties!

There are 2 parts to justifying:

1) First of all, what did you change OR if you’re looking at what someone else did, WHAT CHANGED?

(Did the order change? Did the (  ) change? Has anything been simplified?)

2) What allowed you (or them) to make that change?

(Commutative? Associative? Distributive?)

Example: You’re given (565)(5)(2) but you change it to:  (5)(2)(565) and get quickly (10)(565) = 5650

JUSTIFY! (what did you do to find the answer)

1) You changed the ORDER

2) Commutative Property of Multiplication allows you to change the order

Example: You’re given (565)(5)(2) but you change it to:  (565)[(5)(2)] and get quickly (565)(10) = 5650

JUSTIFY! (what did you do to find the answer)

1) You put in a set of [  ]  

2) Associative Property of Multiplication allows you to either add are take away a set of parentheses

WHY DOES AUNT SALLY DISLIKE PROPERTIES INTENSELY???
BECAUSE PROPERTIES ARE EXCEPTIONS TO HER RULES (ORDER OF OPERATIONS OR PEMDAS)!!!

She’s happy though that sometimes your justification can be ORDER OF OPERATIONS (in other words you just simplified  or did the math in the proper order of PEMDAS)

AGAIN WHY DO WE LOVE PROPERTIES???

WHY SHOULD YOU CARE????
 Because they make the math easier sometimes!
BUT AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!

I.      COMMUTATIVE PROPERTY
PROPERTIES ARE OUR FRIENDS! (mathematically speaking)
YOU CAN ALWAYS DEPEND ON THEM --- THEY HAVE NO COUNTEREXAMPLES!

COMMUTATIVE PROPERTY (works for all multiplication or all addition)
You can SWITCH THE ORDER and still get the same sum or product.
This is the property YOU CAN HEAR because you've switched the order.
a + b = b + a OR ab = ba
Therefore, we say that both sides of the equations have EQUIVALENT (=) EXPRESSIONS
SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???

II.   ASSOCIATIVE PROPERTY
ANOTHER FRIEND!
This friend allows you to GROUP all multiplication or all addition ANYWAY YOU CHOOSE!
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Why? TO MAKE THE MATH EASIER OF COURSE!
This is the property that YOU CAN SEE instead of hearing because you use ( ) but DON'T CHANGE THE ORDER AS IT IS GIVEN.
EXAMPLE: [(543)(5)](2)
Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]
But our friend the Associative Property allows us to simply move the [ ]
[(543)(5)](2) = (543)[(5)(2)] which is so much easier to multiply in your head!!!

TWO MORE FRIENDS: 

III.      THE IDENTITY PROPERTIES OF ADDITION AND MULTIPLICATION
For addition, we know that adding zero to anything will not change the IDENTITY of what you started with: a + 0 = a (what you started with)
0 is known as the ADDITIVE IDENTITY.

For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with: (1)(a) = a (what you started with)
1 is known as the MULTIPLICATIVE IDENTITY.
Sometimes 1 is "incognito" (disguised!)
We use this concept all the time to get EQUIVALENT FRACTIONS.
Say we have 3/4 but we want the denominator to be 12
We multiply both the numerator and the denominator by 3 and get 9/12
We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3
ANYTHING OVER ITSELF = 1 (except zero because dividing by zero is UNDEFINED!)
a + b - c = 1
a + b - c

We also use this property to SIMPLIFY fractions. We "cross cancel" all the parts on the top and the bottom that equal 1 (your parents would say that we are reducing the fraction)

                              6abc = 3bc since both the numerator and denominator can be divided by 2a.

                               2a

IV.  PROPERTIES OF EQUALITY
(these are also called AXIOMS)

REFLEXIVE:
a = a
3 = 3
In words: It looks exactly the same on both sides! (like reflecting in a mirror)
This seems ridiculous, but in Geometry it's used all the time.
I'll show you that in class.

SYMMETRIC:
a = b then b = a
3 + 5 = 8 then 8 = 3 + 5

In words: You can switch the sides of an equation.
We use this all the time to switch the sides if the variable ends up on the right side:
12 = 5y -3
The Symmetric property allows us to switch sides:
5y - 3 = 12

TRANSITIVE:

                        If a = b and b = c then a = c

                        3 + 5 = 8, and 2 + 6 = 8 then 3 + 5 = 2 + 6
                        In words: If 2 things both equal a third thing, then we can just say that the first 2 things are equal.
                        If Jane is 14 years old and Bobby is 14 years old, then we can say that Jane and Bobby are the same age
                        It's like cutting out the "middle man"!

I've got a pattern that will help you recognize the difference between these 3 properties specifically.

If you put these 3 properties in order alphabetically, they'll be in order this way:
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations

SO REMEMBER THIS: R,S,T…1,2,3!

V.      SUBSTITUTION:

Very similar to Transitive

If a = b, then a may be SUBSTITUTED in for b in any other expression.

If 3 + 5 = 8 then 3 + 5 may be substituted for 8 in any other expression:

50 + 8 = 58

50 + (3 + 5) also = 58

In Algebra, we use substitution all the time to substitute a value in for a variable:

3 + n if n = 10

3 + 10 would be an equivalent expression because n = 10 so we can replace n with 10 in the original expression

VI.  INVERSE PROPERTIES:

ADDITIVE INVERSE:

Adding opposites signs of the same term = 0.

This "friend" saves us time when adding a lot of integers together (THAT’S WHY WE SAY “YAY”!)...always look for opposites FIRST and cross them out!

a + (-a) = 0

MULTIPLICATIVE INVERSE:

Multiplying by the reciprocal of a term = 1.

(a)(1/a) = 1

(4/5)(5/4) = 1

(-2)(-1/2) = 1

This friend helps because you can make math easier with fractions by allowing you to cross cancel!

Both inverses are used in equation balancing.

Welcome to Math! Welcome to K101

Our Class Blog

Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day... or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Algebra Honors ( Periods 4 & 7)

CHAPTER 8-5: (2nd part that we skipped) FACTORING 4-TERM POLYNOMIALS

It’s always been called FACTORING BY GROUPING before the Common Core ;)

Today, you'll have 4 TERMS IN YOUR POLYNOMIAL!
You put the polynomial in 2 sets of 2 by using ( )s
Then you factor out the GCF for each set of 2 terms

DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?
Of course not! But for this section of the math book, it will!

What happens if it doesn't work? The polynomial may just not be factorable! (prime)

MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ - 9x² + 4x - 6

NOTICE THAT THERE IS NO GCF OF ALL 4 TERMS!

Factoring by grouping says if there is no GCF of all 4 terms, look and see if there is a GCF of just 2 terms at a time.

Put ( ) around the first 2 terms and the 2nd 2 terms:
(6x³ - 9x² ) + (4x - 6)

Always make sure that there is a + sign in between the two pairs of (   ). If there is a negative sign in front of the 3rd term, keep it inside the 2nd set of (  ) and add a + sign in between.

FACTOR OUT THE GCF FROM EACH SET OF TWO TERMS:
3x² (2x - 3) + 2(2x - 3)

        LOOK AND SEE IF WHAT'S LEFTOVER IN THE (  ) IS THE SAME:

In this case, it was because they were both (2x - 3)

So now this is a GCF of both terms and you can factor that out:
(2x - 3)(3x² + 2)

ALWAYS MAKE SURE THAT IF THERE IS A NEGATIVE ON THE THIRD TERM, PLACE IT IN THE SECOND (  ) AND ALWAYS HAVE A + BETWEEN THE TWO SETS OF (  )

Always ask " Am I done" to MAKE SURE YOU CAN'T FACTOR ANY MORE!

HOMEWORK:

Algebra ( Period 1)

CHAPTER 8-5: (2nd part that we skipped) FACTORING 4-TERM POLYNOMIALS

It’s always been called FACTORING BY GROUPING before the Common Core ;)

Today, you'll have 4 TERMS IN YOUR POLYNOMIAL!
You put the polynomial in 2 sets of 2 by using ( )'s
Then you factor out the GCF for each set of 2 terms

DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?
Of course not! But for this section of the math book, it will!

What happens if it doesn't work? The polynomial may just not be factorable! (prime)

MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ - 9x² + 4x - 6

NOTICE THAT THERE IS NO GCF OF ALL 4 TERMS!

Factoring by grouping says if there is no GCF of all 4 terms, look and see if there is a GCF of just 2 terms at a time.

Put ( ) around the first 2 terms and the 2nd 2 terms:
(6x³ - 9x² ) + (4x - 6)

Always make sure that there is a + sign in between the two pairs of (   ). If there is a negative sign in front of the 3rd term, keep it inside the 2nd set of (  ) and add a + sign in between.

FACTOR OUT THE GCF FROM EACH SET OF TWO TERMS:
3x² (2x - 3) + 2(2x - 3)

        LOOK AND SEE IF WHAT'S LEFTOVER IN THE (  ) IS THE SAME:

In this case, it was because they were both (2x - 3)

So now this is a GCF of both terms and you can factor that out:
(2x - 3)(3x² + 2)

ALWAYS MAKE SURE THAT IF THERE IS A NEGATIVE ON THE THIRD TERM, PLACE IT IN THE SECOND (  ) AND ALWAYS HAVE A + BETWEEN THE TWO SETS OF (  )

Always ask " Am I done" to MAKE SURE YOU CAN'T FACTOR ANY MORE!

HOMEWORK:

Algebra Honors ( Periods 4 & 7)

CHAPTER 8-6: FACTORING TRINOMIALS

We’ll factor them and then put our FULLY FACTORED FORM on the same graph as the SIMPLIFIED FORM….WHAT SHOULD HAPPEN IF WE FACTORED THE TRINOMIAL (quadratic, 2nd degree polynomial) CORRECTLY?

FACTORING TRINOMIALS WITH A POSITIVE SIGN AS THE SECOND SIGN:

You're trying to turn a trinomial back to the two binomials that were multiplied together to get it!
Always check your factoring by FOILing back!

FACTORING TRINOMIALS WITH A 

PLUS SIGN AS THE SECOND SIGN:
I have a simple method of foiling basic trinomials!

1. Set up your (      )(      )

2. When the last sign is positive, then both signs in each of the (     ) are the same!

3. How do you know what the 2 signs are? It's whatever the sign is of the second (middle) term.  Put that sign in both parentheses.

4. To UNFOIL (factor), you will need to find
2 FACTORS that MULTIPLY to the LAST term, and ALSO ADD to the MIDDLE term.
To help you do this, I suggest you use and “X”
Put the product in the top of the X,

the sum in the bottom of the X and

the correct factors on the left and right of the X....

Understand that this is really just an educated guess and check!
  

EXAMPLE:
x² + 8x +15
(       )(       )
THINK: Last sign is + so both signs are the same
THINK: Middle sign is + so both signs are +
(   +   )(   +   )
You know the the "F" in FOIL means that both first terms must be x
( x + )( x + )
Now to get the "L" in FOIL, you need 2 factors whose product is 15
Like 1 and 15, or 3 and 5
But you also need to add to the I and O in FOIL which means that the two factors must add to 8
____ x ____ = 15
____ + ____ = 8
Since 3 + 5 = 8, this must be the two factors that will work:
3 x 5 = 15
3 + 5 = 8
( x + 5 )( x + 3 ) 

(It doesn't matter which factor you put in the first (  )  because they're the same sign, but I always tend to put the larger number in the first parentheses for a reason that you will see tomorrow)
Now FOIL to see if we're right!

NEXT EXAMPLE: LOOKS THE SAME WITH ONE DIFFERENCE...MIDDLE TERM IS NEGATIVE
x² - 8x +15

(        )(        )
THINK: Last sign is + so both signs are the same
THINK: Middle sign is - so both signs are -
(   -   )(   -   )
You know the the "F" in FOIL means that both first terms must be x
( x - )( x - )
Now to get the "L" in FOIL, you need 2 factors whose product is 15
Like 1 and 15, or 3 and 5
But you also need to add to the I and O in FOIL which means that the two factors must add to 8
-___ x -___ = 15
-___ + -___ = -8
Since -3 + -5 = -8, this must be the two factors that will work:
-3 x -5 = 15
-3 + -5 = -8
( x - 5 )( x - 3 ) 

(It doesn't matter which is first because they're the same sign!)
Now FOIL to see if we're right!

I actually just ignore the signs while making the educated guess because I've already put negative signs in both parentheses so I've taken care of the negatives. It's up to you which way you are most comfortable...we'll talk about that in class and you'll try both ways...then YOU make up your mind which way you want to do it and STAY with that method!

LAST EXAMPLE: SAME PROBLEM BUT NOW WITH A y ON THE MIDDLE AND LAST TERMS:
x² - 8xy + 15y²

Simply use the same factorization as above and add the y:
( x - 5y)( x - 3y)

ALWAYS FOIL BACK TO CHECK!!!

I'll also show you "little smile" "big smile" to just check the middle term....that's usually where you make a mistake if there is one....It makes the check step shorter because you're just doing the "O" and "I" in FOIL instead of the entire 4 multiplications.

FACTORING TRINOMIALS WITH A NEGATIVE SIGN AS THE SECOND SIGN:


1. Set up your (       )(       )

2. Look at the SECOND or last sign
If it's negative, then the signs in the (   ) must be different

Why? Because when you multiply integers and get a NEGATIVE product, the only way that will happen is if they are DIFFERENT signs. Remember that the last term is the product of the two LAST terms in FOILing.

3. Now look at the sign of the second term.
It tells you "Who wins," meaning which sign must have the bigger number (absolute value)
Remember that the middle term is the SUM of the "O" and the "I" terms when FOILing. Because these two terms have DIFFERENT signs, when you add them, you actually "subtract" and take the bigger number's sign.

Put that sign in the first parentheses and always put the bigger number in the first parentheses. (Although you can reverse the signs, but then just remember the bigger factor goes with the sign of the 2nd term.

4. To UNFOIL (factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but SUBTRACT to the middle term. (yesterday the factors needed to ADD to the middle)
Or you can still say you're adding, but since they are DIFFERENT signs, you will end up subtracting!
Again, this is still an educated guess and check!

EXAMPLE:
x² + 2x -15
(       )(       )
THINK: Last sign is - so signs are DIFFERENT!
THINK: Middle sign is + so POSITIVE MUST WIN
(   +   )(   -   )
You know the the "F" in FOIL means that both first terms must be x
( x +  )( x -  )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
Like -1 and 15, or -3 and 5, or 1 and -15, or 3 and -5
But since the POSITIVE must win according to the middle term of POSITIVE 2x, you know that the bigger factor must be POSITIVE (so it can win!)
Therefore, your choices are POSITIVE 15 and -1 or POSITIVE 5 and -3
But you also need to add to the I and O in FOIL so pick the two factors

that also add to POSITIVE 2
Actually, I think to myself the two factors that SUBTRACT to 2, since when you add opposite signs, you don't add, you take the difference and take the higher number's sign.
Since -3 + 5 = +2, this must be the two factors that will work.

So adding it would look like this:
+____ x -____ = -15
+____ + -____ = + 2
so
5 x (-3) = -15
5 +( -3) = 2
( x + 5 )( x - 3 )


NEXT EXAMPLE: Same as before except the middle sign is now negative
x² - 2x -15
(      )(      )
THINK: Last sign is - so signs are DIFFERENT!
THINK: Middle sign is - so NEGATIVE MUST WIN
that also add to POSITIVE 2
Actually, I think to myself the two factors that SUBTRACT to 2, since when you add opposite signs, you don't add, you take the difference and take the higher number's sign.
Since -3 + 5 = +2, this must be the two factors that will work.
I set it up like this:
____ x ____ = 15
____ - ____ = 2
so
5 x 3 = 15
5 - 3 = 2

USING ADDING IT WOULD LOOK LIKE THIS:
+____ x -____ = -15
+____ + -____ = + 2
so
5 x (-3) = -15
5 +( -3) = 2
( x + 5 )( x - 3 )

LAST EXAMPLE: Same as the last example but with a “y”:
x² - 2xy -15y²
Same problem as the one before, except now there are 2 variables!
Simply use the same factorization and add the y
( x - 5y )( x + 3y)