Math 6A (Periods 2 & 4)

Tests for Divisibility 5-2

It is important to learn the following divisibility rules:
A number is divisibility by:

2 ... if the ones digit of the number is even
3 ... if the sum of the digits is divisible by three ( add the digits together)
4 ... if the number formed by the last two digits is divisible by by four ( Just LOOK at the last two numbers-- DON"T ADD them!!)
5 ... if the ones digits of the number is a 5 or a 0
6 ... if the number is divisible by both 2 and 3... (or if it is even and divisible by 3)
8 ... if the number formed by the last three digits is divisible by 8. (Like FOUR, just look at the last three digits-- divide them by 8)
9 ... if the sum of the digits is divisible by 9
10 ... if the ones digits of the number is a 0.

You will not need to know the divisibility rules for 7 or 11 but they are interesting...

You can test for divisibility by 7
Let's start with a number 959
Step 1: drop the one's digit so we have 95
Step 2: Subtract twice the ones' digit ( that you dropped) in this case we dropped a 9
so we double that and subtract 18 from 95
or 95-18 = 77. If the results, in the case, 77, is divisible by 7 --- so is the original number 959.
Step 3: If the number you get is still to big.. continue the process until you can determine if your number is divisible by 7.


To test for divisibility by 11
add the alternative digits beginning with the first
so let's try the following
4,378,396
Step 1: Add the alternate digits beginning with the 1st 4 + 7+ 3 + 6 = 20
Step 2: Add alternate digits beginning with the 2nd 3 + 8 + 9 = 20

Step 3: If the difference of the sums is divisible by 11 so is the original number.
In this case, 20-20 = 0 and 0/11= 0 so
4,378,396 is divisible by 11.


A good test for divisibility by 25 would be if the last two digits represent a multiple of 25.

A perfect number is one that is the SUM of all its factors except itself. The smallest perfect number is 6, since 6 = 1 + 2+ 3
The next perfect number is 28 since
28 = 1 + 2 + 4 + 7 + 14
What is the next perfect number?

Algebra Honors ( Period 5 & 6)

Using Factoring to Solve Problems  5-13

Example 1

A decorator plans to place a rug in a 8 m by 12 m room so that a uniform strip of wood flooring around the rug will remain uncovered. How wide will this strip be if the area of the rug is to be half the area of the room?

Let x = the width of the strip

Then 12-2x is the length of the rug and 9 – 2x is the width of the rug

(12-2x)(9-2x) = the area of the rug.

Area of the rug = ½ ( area of the room)

(12-2x)(9-2x) = ½(9∙12)

108 – 42x + 4x² = 54

4x²- 42 + 108 = 54

4x²- 42 -54 = 0

2(2x²- 21 + 27) =0

2(2x -3)(x -9) = 0

using the ZERO Products Property

2x -3 = 0 or x= 3/2

x -9 = 0  x = 9

CHECK: x = 1.5

works but when x = 9

the length 12 – 2x and the width 9-2x are negative! Since a negative length or width is meaningless reject x = 9 as an answer

This show that also the equation has a root that does not check because this equation does not  meet the hidden requirements that the rug have a positive length.  

Example 2
The FORMULA h = rt – 4.9t² is a good approximation of the height (h) in meters of an object t seconds after it is projected upward with an initial speed of r meters per second.

An arrow is shot upward with an initial speed of 34.3 m/s. When will it be at a height of 49 m?

let t = the  number of seconds  after being shot that the arrow is 49 m high.

Let h = the height of the arrow= 49 m

Let r = the initial speed = 34.3 m/s

Substitute in the formula

h = rt – 4.9t²

49 = 34.3t – 4.9t²

4.9t² – 34.3t + 49 = 0  THINK—GCF???

4.9(t² – 7t + 10 ) = 0

4.9(t -2)(t -5) = 0

Using the ZERO PRODUCTS PROPERTY

t = 2 and t = 5

Therefore the arrow is 49 m high both 2 seconds and 5 seconds after being shot… on its way up and on its way down!

Example 3

If a number is added to its square, the results is 56. Find the number

 Let x = the number

x² + x = 56

(x +8)(x - 7) = 0

 x = -8 and x = 7

Find two consecutive positive odd integers whose product is 143

Let x = the first positive odd integer

Let x + 2 = the 2^nd positive odd integer

x(x + 2) = 143

x² + 2x = 143

x² + 2x – 143 = 0

(x + 13)(x – 11) = 0

x = -13 and x = 11

But ask only for the positive integers so reject -13

The two integers are 11 and  13

Example 4

The sum of the squares of two consecutive negative odd integers is 290. Find the integers

let x = the 1^st negative odd integer

let x + 2 = the 2^nd negative odd integer

x² + (x + 2)² = 290

x² + x² + 4x + 4 = 290

2x² + 4x – 286 = 0  THINK  GCF!!!

2(x²+ 2x -143) = 0

2(x + 13)(x -11) = 0

x = -13 and x = 11

But it ask for the negative integers  so reject x = 11

The two negative integers are -13 and -11

Math 6A (Periods 2 & 4)

Finding Factors and Multiples 5-1


You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 60. A number is said to be divisible by its whole numbered factors.

To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.

We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6

When you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!

A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first five multiples of 7 are
0, 7, 14, 21, 28
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21; 4(7) = 28
... and put in set notation it would be
{0, 7, 14 ,21, 28}



If you were to ask for the first four NON-ZERO Multiples of 6
the answer would be 6, 12, 18, 24.. and in set notation
{ 6, 12, 18, 24}
Whereas the first four multiples of 6 ( you would need to include 0)
{0, 6, 12, 18}



Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!

Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number

What number is a factor of every number? ONE
Is every number a factor of itself? YES
What is ( are) the only multilpe(s) of 0? 0
How many numbers have 0 as a factor? only one number What number(s)? ZERO



The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.

Algebra Honors (periods 5 & 6)

Solving Equations by Factoring Section 5-12

a⋅0 = 0
and if a = 0 or b = 0
then we know that ab= 0
This is an if, then statement
conversely
if ab = 0 then either a= 0 or b = 0
THis Zero Products Property helps us solve equations.

(x +2)(x -5) = 0
either x + 2 must equal zero or x - 5 must equal zero
so set each expression equal to zero and solve
x + 2 = 0
x= -2
and x-5 = 0
x = 5

{-2. 5}

5m(m-3)(m-4) = 0
now you have three expressions so set each of them to zero
5m = 0 so m = 0
m-3 = 0 so m=3
m-4 = 0 so m=4
{0,3,4}

What happens with
3x²+ x = 2
It isn't the 2 products property but the ZERO products property so set the expression equal to ZERO
3x²+ x -2 = 0
Now factor
(x+1)(3x -2) = 0
set each of these equal to zero
x + 1 = 0 x = -1
3x -2 = 0 so x = 2/3

{-1, 2/3}

10x³ - 15x² = 0
factor
5x²(2x -3) = 0
again set each equal to zero
5x² = 0 so x = 0
and
2x -3 = 0 so x = 3/2

{0, 3/2}

polynomial equation named by the term of highes degree

ax + b = 0 linear equation

ax² + bx + c = 0 quadratic equations

ax³ + bx² + cx + d = 0 cubic equation

2x² + 5x = 12
becomes
2x² +5x - 12 = 0
(x + 4)(2x-3) = 0
so x = -4 and x = 3/2
{-4, 3/2}

18y³ + 8y + 24y² = 0
Rearrange first
18y³+ 24y ² + 8y = 0
Then factor the GCF
2y(9y² +12y +4) = 0
WAIT--> its a PERFECT trinomial SQ
2y(3y +2)² = 0
2y = 0 so y = 0
and 3y + 2 = 0 so y = -2/3

-2/3 is a double or multiple root but you only list it once in solution set.
That is,
{-2/3, 0}

y = x² + x - 12
solve for the roots means you set this quadratic equal to ZERO
so
x² + x - 12 = 0
(x+4)(x -3) = 0

Math 6A ( Periods 2 & 4)

Dividing Decimals 3-9 cont'd

For word Problems use the 5 step plan found on Page 18 of our textbook

296.06 ÷ (18.7 + 3.9)
Following Aunt Sally ( or PEMDAS... remember our singing...
we do the operation inside the hugs!! ( )using a sidebar
18.7 + 3.9 make sure to stack them lining up the decimals and you will get 22.6

296.06 ÷ 22.6

When dividing by a decimal remember the rule from yesterday, multiply the divisor ( 22.6) by a power of ten which makes it a natural number

then use that same power of ten and multiply the dividend,
WHen you divide you have
2960.6 ÷ 226
Please do that problem and your quotient should be 13.1

(47.1 - 16.9) ÷ (21.9 -6.8)
Again you need to do the operations inside the ( ) first. Using a side bar and lining up the decimals
47.1 - 16.9 = 30.2
and 21.9 - 6.8 = 15.1
Just take a look at those two numbers and you will notice a relationship!!
30.2 ÷ 15.1
BUT... practice your division skills and confirm what you can tell...
30.2 ÷ 15.1 = 2



At an average rate of 55 km/hour how long will it take to drive 225 km to the nearest tenth of an hour?

d = rt
What must we find and what are the clues? Well, how long... is usually time and the fact that we need to round to the nearest tenth of an hour indicates we are finding TIME as well.
So what is the distance? 225 km and what is the rate? 55 km/h
so plug into the formula
225= 55t
Now, how do we solve this one step problem?

divide both sides by 55
225/55 = 55t/55

do the division as a side bar

225/55 ≈ 4.09 so
t ≈ 4.1
and the answer is 4.1 hour

Math 6A (Periods 2 &4)

Dividing Decimals 3-9

According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.

Check out our textbook for some examples!!

When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.

Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!

the quotient is 0.3431 which rounds to 0.343


To divide one decimal by another

Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number


Divide the new dividend by the new divisor

Check by multiplying the quotient and the divisor.

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

Math 6A (Periods 2 & 4)

Rounding 3-5 (cont'd)

Round the following number to the designated place value:
509.690285

tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7

hundredths
509.690285
becomes 509.69

hundred-thousandths
509.690285
becomes
509.69029

tens
509.690285
becomes
510

(a) What is the least whole number that satisfies the following condition?

(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so

(a) 515
(b) 524

A whole number rounded to the nearest ten is 650
(a) 645
(b) 654

A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?

A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49

A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99

Algebra Honors (Periods 5 & 6)

Factoring Pattern for x² + bx+ c, c positive 5-7

In this lesson we will be factoring trinomials that can be factored as a product of ( x +r)(x + s)
where r and s are both positive OR both negative integers.

x² + ( r + s)x + rs  

(x +3)(x+5) = x² + 8x + 15

(x – 6)(x -4) = x² -10x + 24

where -10 is the sum of -6 and -4

and

24 is the product of -6 and -4

Our book suggests that you list all the pairs of integral factors whose products equal the constant term Then, find the pair of integral factors whose SUM equals the coefficient of the linear term (remember your new vocab)

For factoring x² + bx + c, where c is positive  you only need to consider factors WITH THE SAME SIGNS as the linear term!!

In class, I showed the diamond method of calculating products and sums… and even recommended an app to practice!!
Here is the link for the iphone,  ipad app…

y² + 14y + 40

Since the linear term ( +14y) is positive you know to set up two sets of HUGS with + in the middle

(  +  )(  +  )

then you can add the single y’s since y² = y·y

(y + )(y + )

Now either list all the integral factors or use the diamond method and you discover that 10 and 4 are the two factors that multiply to 40 AND also sum to 14

(y + 10 )(y + 4)

To check if you are accurate, use FOIL or Fireworks… or the BOX method and see if you get back to the original problem!

y2 – 11y + 18

This time notice that the linear term is -11y so you will be looking for a pair of numbers whose product will be positive  ( so both need to be negative)

Set  up your  HUGS  similarly—EXCEPT both signs need to be NEGATIVE

(y -   )( y -  )

Since -11 is negative, think of the negative factors of 18 using the book’s method or the diamond method

hmmm… -9 and -2 work

(y - 9  )( y -  2)

Again To check if you are accurate, use FOIL or Fireworks… or the BOX method and see if you get back to the original problem!

A polynomial that cannot be expressed as a product of polynomials of lower degree is said to be irreducible. An irreducible polynomial with integral coefficients whose greatest monomial factor is 1 is a PRIME POLYNOMIAL.

Factor x² -10x + 14

Setting up your HUGS   you start to think what two factors multiply to 14 and SUM to 10… hmmm. NOTHING…

Therefore x² -10x + 14  cannot be factored and it is a prime polynomial

Find all the integral values of k for which the trinomial can be factored

x² + kx + 28

28 can be factored as a product—using  a T chart

list all the factors

(1)(28) (2)(14) (4)(7)

Taking the corresponding sums you get 29, 16 and 11

BUT… remember you can also have the negatives here

so the values of k can be ± 29,   ± 16,   ± 11

{-29, -16, -11, 11, 16, 29}