Algebra Honors (Period 6 & 7)

Differences of Two Squares 5-5

(a + b)(a-b) = a² - b²
(a+b) is the sum of 2 numbers
(a-b) is the difference of 2 numbers
= ( first#)² - ( 2nd #) ²

( y -7)( y + 7) = y² - 49
We did the box method to prove this.

(4s + 5t) (4s - 5t)
16s² - 25t²

(7p + 5q)(7p-5q) = 49p² - 25q²
But then we looks at
(7p+5q)(7p+5q) that isn't the difference of two squares that is
49p² + 70pq + 25q²
So let's look at the difference of TWO Squares:

b² -36
So that is ( b + 6)(b -6)

m² - 25
(m + 5)(m -5)

64u² - 25v²
(8u + 5v)(8u -5v)

1 - 16a²

(1+4a)(1- 4a)

But what about 1- 16a⁴

( 1 + 4a²)(1 - 4a²) but we are NOT finished factoring because
(1 - 4a²) is still a difference of two squares so it becomes

( 1 + 4a²)(1 + 2a)(1 - 2a)

t⁵ - 20t³ + 64t

Factor out the GCF first

t(t⁴ - 20t² + 64)
t(t² -16)(t² -4)
YIKES... we have two Difference of Two Squares here...

t(t +4)(t-4)(t +2)(t -2)



81n² - 121
(9n +11)(9n -11)
3n⁵ - 48 n³


Factor out the GCF

3n³ (n² - 16)
3n³(n +4)(n-4)

50r⁸ - 32 r²

Factor out the GCF
2r²(25r⁶ - 16)

2r²(5r³ +4)(5³ -4)

u² - ( u -5) ²
think a² - b ² = (a + b)(a -b)

so
u² - ( u -5) ² =
[u + (u-5)][u - (u-5)]
(2u -5)(5)
= 5(2u-5)

t² - (t-1)²

[t +( t+1)]{t-(t-1)]
2t-1(+1)
=2t -1


What about x²ⁿ - y ⁶ where n is a positive integer

well that really equals
(xⁿ)² - (y³)²
so
(xⁿ + y³)(xⁿ - y³)

x²ⁿ - 25
(xⁿ + 5)(xⁿ - 5)

a⁴ⁿ - 81b⁴ⁿ
(a²ⁿ + 9b²ⁿ)(a²ⁿ - 9b²ⁿ)
= (a²ⁿ + 9b²ⁿ)(aⁿ + 3bⁿ)(aⁿ - 3bⁿ)


When multiplying to numbers such as (57)(63)
think
(60-3)(60 +3)
then the problem becomes so much easier

3600 - 9 = 3591 DONE!!!

(53)(47) = (50 +3)( 50-3)
2500 - 9 = 2491

Math 6 Honors ( Periods 1, 2, & 3)

Rounding 3-5

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 (Period 6 & 7)

Multiplying Binomials Mental 5-4

Look at
(3x - 4)(2x+5)

remember FOIL
First terms (3x)(2x)
Outer terms (3x)(5)
Inner Terms (-4)(2x)
Last Terms (-4)(5)

6x² + 15x -8x -20
6x² + 7x - 20
Or use the box method as we have done in class
This is a quadratic polynomial
The quadratic term is a term of degree two

Remember a linear term has a term of degree 1 such as y = 3x + 5

6x² + 7x - 20

The 6x² is the quadratic term
the +7x is the linear term
and the - 20 is the constant term

(x +1)(x +3) = x² + 4x + 3

(y + 2)( y + 5) = y² + 7y + 10
( t -2)( t -3) = t² -5t + 6

( u -4)(u -1) = u² - 5u + 4

What about
( u-4)(u +1) = u ² -3u -4

See the difference between the two?

(7 - k)(4 -k)

28 - 11k + k²


r + 3)(5 - 5)
r² - 25 - 15



(3x - 5y)(4x + y)
12x² - 17xy - 5y²

a + 2b)(a-b)

careful....
a² + ab - 2b²

n(n-3)(2n+1)
first distribute the n
(n² -3n)(2n +1)
2n³ - 5n² - 3n

Solve for
(x-4)(x +9) = (x +5)(x -3)
x² + 5x - 36 = x² + 2x -15
5x - 36 = 2x - 15
3x = 21

x = 7
or in solution set notation {7}

Math 6 Honors ( Periods 1, 2, & 3)

Comparing Decimals 3-4



In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:


1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.




2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .




3. 0.83 and 0.833



To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833



0.83 = 0.830 Then compare



0.830 and 0.833 since 0 <3

Then 0.830 < 0.833.



Write in order from least to greatest



4.164, 4.16, 4.163, 4.1



First, express each number to the same number of decimal places

Then compare. 4.164, 4.160, 4.163, 4.100




The order of the numbers from least to greatest is


4.1, 4.16, 4.163, 4.164

Math 6 Honors ( Periods 1, 2, & 3)

We add the following bit of notes today:
looking at the powers of 10 we noticed that
10² ⋅ 10³ = 10⁵
and
10⁸ ⋅10⁶ =10¹⁴
so could we write a rule for any exponent values a and b?
YES, we decided:
10^a ⋅10^b = 10^a+b


Remember how we proved that any number to he zero power was equal to 1
or a⁰ = 1

Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3
or 3⁴⁺² = 3 ⁶
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract

3⁴ /3² = 3^4-2 =3²

What would happen if we had
3² / 3⁴ ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3

Which would be
1
3²

or 1/3²

but you can write that as 3^-2
We just subtract-- using the same rule.


Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN

Math 6 Honors ( Periods 1, 2, & 3)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0.1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.

Remember how we proved that any number to he zero power was equal to 1
or a⁰ = 1

Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3
or 3⁴⁺² = 3 ⁶
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract

3⁴ /3² = 3^4-2 =3²

What would happen if we had
3² / 3⁴ ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3

Which would be
1
3²

or 1/3²

but you can write that as 3^-2
We just subtract-- using the same rule.


Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN

SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

As with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point