Algebra Honors ( Period 4)

Factoring Trinomials 8-6

We  will factor them and then put our fully factored form on the same graph as the simplified form—What do you think will happen if we factored the trinomial (quadratic, 2^nd degree polynomial) correctly?

Factoring Trinominals with a POSITIVE sign as the SECOND SIGN

You are trying to turn a trinomial back to the two binomials that were multiplied together to get it!

Always check your factoring by FOILing back!

There is a simple method for foiling basic trinomials

1) set up your two sets of  {{HUGS}  (    ) (    )
2) When the last sign is positive then BOTH SIGNS in each of the {{HUGS}} are the SAME—they could be Both POSITIVE or they could be BOTH NEGATIVE!!
3) How do you know what the 2 signs are? It is whatever the SIGN is of the MIDDLE ( SECOND) term!! PUT THAT SIGN in both {{HUGS}} 
4) TO UNFOIL ( Factor) you will need to find 2 factors that MULTIPLY to the Last Term & ALSO ADD to the Middle Term
To help you do this I suggest you use an X Put the product in the top of the X and the sum in the bottom of the X . Then figure out the correct factors on the left and right of the X. It gets easier with practice! I promise

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

EXAMPLE 1
x² + 8x + 15

(   )(   )

Two set of {{HUGS}} are set up
THINK: Last sign is + so both of the signs inside the parentheses are the same
THINK: Middle sign is + so both of the signs are  +
( + )( + )

You know that the “F” in FOIL means that both first terms must be x so go ahead an place the x in each
(x + )(x + )
You need to get the “L” in FOIL You need 2 factors whose product is 15 Like 1 and 15 or 2 and 5
If you use the “X” from above  YOU can see it!

But if you are having difficulty realize you need to add to the I and O in FOIL which means that the two factors must add to 8

___ ∙ ____ = 15

____ + ___ = 8

Since 3 + 5 = 8 this must be the two factors that work

3 ∙ 5= 15
3 + 5 = 8

(x + 5)(x + 3)

In this case it doesn’t matter which factor you put in the first set of  {{HUGS}} because they are the same sign. (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 are right!!

EXAMPLE 2
x² - 8x + 15
Looks the same with one difference—the middle term is negative

(   )(   )

Two set of {{HUGS}} are set up

THINK: Last sign is + so both of the signs inside the parentheses are the same
THINK: Middle sign is - so both of the signs are  -
( - )( - )

You need to get the “L” in FOIL You need 2 factors whose product is 15 Like 1 and 15 or 2 and 5
If you use the “X” from above  YOU can see it!
But if you are having difficulty realize you need to add to the I and O in FOIL which means that the two factors must add to 8

-___ ∙ -____ = 15

-____ + -___ = -8

Since -3 + -5 = -8 this must be the two factors that work

-3 ∙ -5= 15
-3 + -5 = -8

(x - 5)(x - 3)

I actually just ignore the signs which making an educated guess because I have already put the negative signs in both parentheses so I have taken care of the negatives. It is up to you which way you are most comfortable… But once you make yup your mind--> stick with that method

EXAMPLE 3
Same problem but now with a “y” on the middle and last terms
Looks the same with one difference—the middle term is negative  Simply use the same factorization as above and add the y

x² - 8xy + 15y²

Simply use the same factorization as above and add the y

(x- 5y )(x – 3y)
Always FOIL back to check!
Remember the Little nose and the Big Smile—to check the middle term—that is usually were students make a mistake. It is the O and I in FOIL

Factoring Trinomials with a Negative Sign as the Second Term
This is a bit more complicated

1) set up your two sets of  {{HUGS}}  (    ) (    )
2) Look at that last sign, If it is NEGATIVE, then the signs in the {{HUGS}} must be DIFFERENT. Why? Because in order to get a negative when you multiply integers—one of them needs to be negative and the other must be positive. 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 or which has the greater 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 take the difference (subtract) and take the “bigger number’s”  sign.
PUT THAT SIGN IN THE FIRST PARENTHESES and always put the “bigger number” in the first set of {{HUGS}}
4) TO UNFOIL ( Factor) you will need to find 2 factors that MULTIPLY to the Last Term & ALSO SUBTRACT to the Middle Term  (YOU CAN STILL SAY YOU ARE ADDING BUT SINCE THEY ARE DIFFERENT SIGNS YOU ARE TAKING THE DIFFERENCE)

To help you do this I suggest you use an X Put the PRODUCT in the top of the X and the DIFFERENCE in the bottom of the X . Then figure out the correct factors on the left and right of the X. It gets easier with practice! I promise

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

EXAMPLE 4
x²+ 2x – 15

(   )(   )

Two set of {{HUGS}} are set up
THINK: Last sign is - so both of the signs inside the parentheses are DIFFERENT
THINK: Middle sign is + so the POSITIVE WINS
( + )( - )

You know 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

-1 and 15, 1 and -15 or -3 and 5 or 3 and -5

But, since the POSITIVE must win, according to the POSITIVE TERM of 2x, you  know that the bigger factor must be POSITIVE ( so it can win)
so either + 15 and -1 or +5 and -3 are the only choices
But, you also need to add to the I and the O in FOIL so pick the two factors that also ADD to POSITIVE 2
Therefore it has to be +5 and -3. 

+___ ∙ -____ = 15

+___ + -___ = 2

+5 ∙ -3 = 15
+5 + -3 = 2

so    (x +5 )(x - 3)

EXAMPLE 5
x²-  2x – 15
That’s the same as before EXCEPT the middle sign is now negative

(   )(   )

Two set of {{HUGS}} are set up

THINK: Last sign is - so both of the signs inside the parentheses are DIFFERENT
THINK: Middle sign is - so the NEGATIVE WINS
Its exactly the same but this time you need to get to a difference of -2
(x -5 )(x +3)

EXAMPLE 6
Same as the last example but this time with a “y”

x²-  2xy – 15y²

Same problem except that there are two variables—make sure to add the Y at the end—or set it up right from the start…(   )(   )

Two set of {{HUGS}} are set up

(x y)(x y)

Use the same technique

(x - 5y)(x + 3y)

Math 8 ( Period 1)

Use the Pythagorean Theorem 5-6

We use the Pythagorean Theorem to solve many real word problems:

The Pythagorean Theorem formula: a² + b² = c²

For example, the height of a ladder leaning on a building (that’s a right triangle)
The height of a tree if you know the height of the ladder leaning against it and how far the ladder is from the base of the tree

The length of a wire needed to support a flag pole if you know its height.

The height of a kite if you know the length of its string and how far it is away from where you are standing.

As long as you can visualize the problem as a RIGHT TRIANGLE you can use the formula to solve for an unknown measurement.

Example 1: How high a 10 foot ladder reaches on a house
A 10 ft ladder is placed on a house  5 ft away from the base of the house.
Find how high up the house the ladder reaches
The ladder makes a right triangle with the ground being one leg, the house being the other and the ladder is actually the hypotenuse.
You need to find the distance on the house so you are finding one of th legs in the right triangle.

Example 2: You are flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you area standing 9 feet away from where the kite is in the air.  The string represents the hypotenuse. You know one leg is the height in the air ( 13 ft) and the other leg is how far on the ground you are standing away from where the kite  is flying ( 9ft) This time you need to find the hypotenuse

The Pythagorean Theorem formula: a² + b² = c²