Algebra Honors (Periods 5 & 6)

Factoring by Grouping 5-10


5(a -3) - 2a (3 -a)

a-3 and 3-a are OPPOSITES
so we could write 3-a as -(-3 +a) or -(a -3)
sp we have
5(a-3) -2a [-(a-3)]
which is really
5(a-3) + 2a(a-3)
wait... look... OMG they both have a-3
so
(a-3)(5 + 2a)

What about
2ab-6ac + 3b -9c

What can you combine...
some saw the following:

(2ab -6ac) + 3b -9c)
then
2a(b-3c) + 3( b-3c)
(b -3c)(2a + 3)

BUT others look at 2ab-6ac + 3b -9c and saw
2ab +3b -6ac -9c
which lead them to
(2ab + 3b) + (-6ac -9c)
b(2a +3) -3c(2a +3)
(2a +3)(b-3c)
wait that's the same!!
Hooray

What about 4p² -4q² +4qr -r²
First look carefully and you will see

4p² -4q² +4qr -r²
That's a trinomial square OMG

so isn't that
4p² - ( 2q -r)²

BUT WAIT look at

4p² - ( 2q -r)² That's the
Difference of Two Squares
Which becomes
(2p + 2q -r)(2p -2q +r)

Algebra Honors (Periods 5 & 6)

Factoring Pattern for ax² + bx + c Section 5-9

When a > 1
We used a different method than what is taught in the book. I showed you X box

2x² + 7x -9
Multiply the 2 and the 9
put eighteen in the box
Your controllers are
2x² and -9
THen using a T chart find the factors of 19 such that the difference is 7x
we found that +9x and -2x worked

so
2x² +9x -2x -9
Then separate them in groups of 2
such that


(2x² +9x) + (-2x -9)

Then realize you can factor a - from the second pair

(2x² +9x) - (2x + 9)
Then wht is the GCF in each of the hugs( )
x(2x +9) -1(2x +9)
look they both have 2x + 9
:)
(2x +9)(x-1)
But what if you said -2x + 9x instead to make the +7x in the middle
Look what happens
(2x² -2x) + (9x -9)
now, factor te GCF of each
2x(x -1) + 9(x -1)
now they both have x -1
(x-1)(2x +9)
SAME RESULTS!!

14x² -17x +5
remember the second sign tells us that the numbers are the same and the first sign tells us that they are BOTH negative

create your X BOX with the product of 14 and 5 in it
70

Place your controllers on either side

14x² and + 5

Now do your T Chart for 70
You will need two numbers whose product is 70 and whose sum is 17
that's 7 and 10

14x² -7x -10x + 5

Now group in pairs

(14x² -7x) + (-10x + 5)
which becomes

(14x² -7x) - (10x - 5)

FACTOR each
7x(2x -1) - 5(2x-1)
(2x-1)(7x-5)

10 + 11x - 6x ²

sometimes its better to arrange by decreasing degree so this becomes

- 6x ² +11x + 10

now factor out the -1 from each terms


- (6x ² - 11x - 10)

Se up your X BOX with the product of your two controllers :)
60 We discover that +4x and -15x are the two factors

-1(6x ² +4x - 15x - 10)

-1[(6x ² +4x) + (- 15x - 10)]
-1[6x ² +4x) - (15x +10)
-1[2x(3x +2) -5(3x+2)]
-(3x+2)(2x-5)


If you had worked it out as
10 + 11x -6x² you would have ended up factoring
(5 -2x)(2 + 3x)
and we all know that
5 -2x = -(2x-5) Right ?


Next, we looked at the book and the example of
5a² -ab - 22b²
We discussed the books instructions to test the possibilities and decided that the X BOX method was much better.... I need to check out hotmath.com... did you????

5a² -ab - 22b² Using X BOX method we have 110 in the box and the controllers are
5a² and - 22b²
What two factors will multiply to 110 but have the difference -1?
Why 10 and 11

5a² +10ab -11ab - 22b²

separate and we get
(5a² +10ab) + (-11ab - 22b²)
( 5a² +10ab) - (11ab + 22b²)

5a(a + 2b) -11b(a + 2b)
(a + 2b)(5a - 11b)

Math 6A (Periods 2 & 4)

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000

10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620

762 X 10 = 7620

4931 X 10 = 49,310


10⁴ = 10⋅10⋅10⋅10 = 10,000

2.63874 X 10⁴ = 26,387.4

To multiply a number by the nth power of ten--> move the decimal n places to the right.

0.0047 multiply by 100 = 0.47
0.0047 multiply by 1000 = 4.7

3.1 ÷ 10⁴ = 0.00031


10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the n^th power of ten, move the decimal point n places to the right.



When we move a decimal point to the left, we are actually dividing by a power of ten.


Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

2386 ÷ 10³ = 2.386

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors

(1) a number greater than or equal to 1, but less than 10,

AND

(2) a power of ten

are said to be written in scientific notation.

We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation



4.592 x 10⁹ (Yes, you get to use the × symbol for multiplication .. but only for this!!



Way to write very large numbers AND very small numbers

Numbers expressed as products of a number greater than or equal to 1 BUT less than 10, AND a power of ten are called Scientific Notation.

Two Factors
91) 1≤ n < 10 (2) Power of 10 4,592,000,000 becomes 4.592 X 10⁹
moved the decimal 9 places so we must multiply our number by a power of 10⁹

98,000,000 = 9.8 X 10⁷

320,000 = 3.2 X 10⁵

What if I give you 7.04 X 10⁸ and ask you to put it back into STANDARD NOTATION:

704,000,000.

0.0031 = 3.1 X 10^-3
It isn't a negative number its just a very tiny number

1≤ n < 10 0.16 becomes 1.6 x 10 ^-1

Math 6High ( Period 3)

Using a Least Common Denominator 3.2 

In chapter 2 we learned to find the LCM of two or more whole numbers. When you add or subtract fractions with different denominators,  a convenient denominator is the least common denominator… which is really  the Least Common Multiple of the Denominators… but it is simply called the LCD

Add:
3/8 + 5/ 12

Using the method we talked about yesterday, a common denominator is the product of 8 and 12

8(12) = 96

so we would do the following:

3(12) + 5(40)
96

36+ 40
96

76/96 

but we can simplify this. 

Using the GCF  ( OMG- we use both LCM and the GCF  to simplify fractions!!)

76/96 = 19/24

Using the LCM of 8 and 12 (now—how do we find that ?... Oh yeah.. use the BOX method or inverted division to get the prime factorization of each) So the LCM(8.12) = 24

Now rewrite the fractions

I taught you to stack them ( STACK ‘EM}

3/8 ·3/3= 9/24

5/12∙2/2 = 10/24

No you can add the numerators

9 +10
24

Wow—We arrived at the same solution!

Using the LCD reduces the amount of simplifying you need to do!

But what happens when you need to add three fractions with three different denominators? You can’t use the BOX method of finding the LCM… You need to find the LCM by… using

ALL THE FACTORS TO THEIR GREATEST POWERS… 

ALL THE FACTORS TO THEIR GREATEST POWERS!

1/3 + ¼ + 1/8  Yikes… what to do?

First try to see if one of the denominators is a multiple of the other… you now only need to find the LCD of the that fraction and the one that isn’t a multiply or factor . 

In this case,  8 is a MULTIPLY of 4 so I don’t need to worry about 4

I need to find the LCM  of 3 and 8.. Now that’s easy

LCM(3,8) is their product 24

So change all of the denominators to 24

1/3 = 8/24

¼ = 6/24

1/8 = 3/24

And add

23/24

Evaluating a Variable expression

Evaluate x – y + z  when  x = 9/10  y = 3/4 and z = 1/3

x- y + z = 9/10 – 3/4  + 1/3

Rewrite using the LCD

9/10 ·6/6 = 54/60

3/4·15/15 = 45/60

1/3·20/20 = 20/60

54 – 45 + 20
60

29/60

Algebra Honors (Periods 5 & 6)

 Factoring Pattern for x² + bx+ c, c negative 5-8

Goal- to factor quadratic trinomials whose quadratic coefficient is 1 and whose constant is negative

The method used in this lesson is  very similar to that used to factor x² +bx + c , c is  positive, except instead of the sum of the two factors you  find their difference. Remember  with x² +bx + c , c is positive, you find two numbers whose product is c and whose sum is b.

This time find two numbers whose product is c (which is negative)—so ONE of the TWO factors must be negative. You will have either  (x + )(x - )  or  (x - )(x + ) Since c is negative, one of the two factors MUST be negative.

The first sign in x² + bx + c , c is negative determines “Who wins!”  Let’s rewrite x² +bx + c , c is negative as either   x² +bx - c ,  or x² -bx - c   to see how this works.

We started with x² –x – 20
Set up your hugs…. (x -  ) (x +  )… with the winning sign going in the first set of hugs

Then using the X method find two numbers whose product is 20 and whose difference is 1 ( and in this case actually -1) We found 5 and 4 works and the 5 must be negative to get  -1

so (x -5)(x + 4)

If the quadratic was x²+x-20 you would still have the same factors 5 and 4 but this time the difference is +1 so you would have ( x +5)(x -4)

How about x² + 29a – 30

The factoring pattern is ( x + )(x -  )
Use the X method and find two numbers whose product is 30 and whose difference is 29
We find it has to be 30 and 1  (x+30)(x -1)

To check just FOIL, FireWorks, use the BOX method or just double-distribute to get back to where you started!!

x²-4kx +12k²

This time we have another variable on the last two terms so the factoring pattern starts out as
(x- _k)(x+ _k)
But again we just need to find two numbers whose product is 12 and the difference is 4
6 and 2 work so its ( x -6k)(x + 2k)

Find all the integral values for k for which the given polynomial can be factored.

c²-kc-20

For this exercise, set up a T chart with all the factors that multiply to 20

we found 1 and 20, 2 and 10, and 4 and 5.  Now taking their differences, we find that ± 19 ± 8 ± 1 all work.

Find two negative values for k  for which the given polynomial can be factored. (There are many possibilities—the class found several)

y² + 4y  + k

We found -5, -12, -77,  45, -21, … and the huge set of numbers Jeffrey King found… what were those numbers?

Math 6 High ( Period 3)

Adding & Subtracting Fractions 3.1 

To add fractions with common denominators à just add their numerators ad write this sum over the denominator.

To subtract fractions with common denominators à just subtract their numerators and write the difference over the denominator

1/7 + 3/ 7 = 4/7

7/10 – 3/1-0 = 4/10 = 2/5

In the 2^nd example we found 4/10 to be our answer but then we could simplify… Always simplify your answer.

We then drew the picture on Page 111 of the stack of books and found the height of the stack.

3/8 + 7/8 = 10/8  which is an improper fraction so we changed it to a mixed number 1 ¼  So the books were stacked  1 ¼ in

To add and subtract fractions with different denominators first rewrite the fractions so that they have the SAME denominator. Then use the rules above.

You can use ANY COMMON DENOMINATOR ( the LCD is the Best—but we will discuss that tomorrow in detail)

a/b + c/d = (ad + bc)/bd

If you cannot figure out a common denominator—one of the choices is always to multiply the denominators. That product will always be one of the common denominators!

¼ + 2/ 5 =

1(5) +4(2)
20

13/20