It is all about 1.REAL NUMBERS 2.PROPERTIES OF REAL NUMBERS 3.FACTORING 4.RATIONAL EXPRESSION

4. Real Numbers include all the rational
and irrational numbers.
Real numbers are denoted by the
letter R.
Real numbers consist of the natural
numbers, whole numbers, integers,
rational, and irrational numbers.

5. Natural numbers, whole numbers, integers,
decimal numbers, rational numbers, and
irrational numbers are the examples of real
numbers.
Natural Numbers = {1, 2, 3,...}
Whole Numbers = {0, 1, 2, 3,...}
Integers = {..., -2, -1, 0, 1, 2,...}
, 10.3, 0.6, , , 3.46466466646666..., , are few
more examples.

6. A Rational Number is a real number written as a
ratio of integers with a non-zero denominator.
Rational numbers are indicated by the symbol Q.
Rational number is written in p/q form, where p and
q are integers and q is a non-zero denominator.
All the repeating or terminating decimal numbers
are rational numbers.
Rational numbers are the subset of real numbers.

7. 3/5, 10.3, 0.6 ,12/5, 3/4, - All these are
examples of rational numbers as they
terminate.

8. Irrational numbers are real numbers that
cannot be expressed as fractions, terminating
decimals, or repeating decimals.
Any real number that cannot be expressed as a
ratio of integers.
Are real numbers that cannot be represented
as terminating or repeating decimals.
Fractions.

9. ℮, ,√ 10 are few examples of irrational
numbers.

10. Commonly known as a "whole number", is a
number that can be written without a fractional
component. For example, 21, 4, and −2048 are
integers, while 9.75, 5½, and √2 are not.
Positive or negative numbers or zero.
Integers are the set of whole numbers and their
opposites.
{. . . -3, -2, -1, 0, 1, 2, 3 . . . } is the set of integers.

11. -12,315
733,751
-10
121
56
-89,100
26
1
-75

12. The non integer numbers are defined as
the numbers as the decimal numbers,
fractional number, mixed numbers, etc.

14. The distributive property comes into play when
an expression involves both addition and
multiplication. A longer name for it is, "the
distributive property of multiplication over
addition." It tells us that if a term is multiplied by
terms in parenthesis, we need to "distribute"
the multiplication over all the terms inside.
EXAMPLE:
2x(5 + y) = 10x + 2xy

15. The commutative property of addition says that we
can add numbers in any order. The commutative
property of multiplication is very similar. It says that
we can multiply numbers in any order we want
without changing the result.
Addition
5a + 4 = 4 + 5a
Multiplication
3 x 8 x 5b = 5b x 3 x 8

16. Both addition and multiplication can actually be
done with two numbers at a time. So if there are
more numbers in the expression, how do we
decide which two to "associate" first? The
associative property of addition tells us that we can
group numbers in a sum in any way we want and
still get the same answer. The associative property
of multiplication tells us that we can group
numbers in a product in any way we want and still
get the same answer.
Addition
(4x + 2x) + 7x = 4x + (2x + 7x)
Multiplication
2x2(3y) = 3y(2x2)

17. The identity property for addition tells us that zero
added to any number is the number itself. Zero is
called the "additive identity." The identity property
for multiplication tells us that the number 1
multiplied times any number gives the number
itself. The number 1 is called the "multiplicative
identity."
Addition
5y + 0 = 5y
Multiplication
2c × 1 = 2c

18. If you add real numbers the answer is also real.
ADDITION OF CLOSURE PROPERTY
The problem 3 + 6 = 9 demonstrates the closure
property of real number addition.
Observe that the addends and the sum are real
numbers.
The closure property of real number addition
states that when we add real numbers to other
real numbers the result is also real.
In the example above, 3, 6, and 9 are real
numbers

19. If you multiply real numbers the answer is also
real.
MULTIPLICATION OF CLOSURE PROPERTY
The problem 5 × 8 = 40 demonstrates the closure
property of real number multiplication.
Observe that the factors and the product are real
numbers.
The closure property of real number multiplication
states that when we multiply real numbers with other
real numbers the result is also real.
In the example above, 5, 8, and 40 are real numbers.

20. Inverse Properties state that when a number is
combined with its inverse, it is equal to its
identity.
There are two types of inverses of a number:
Additive Inverse and Multiplicative Inverse.
`- a` is said to be additive inverse of `a` if
a + (- a) = 0.
`1a` is said to be multiplicative Inverse of `a` if
ax 1a=1 .

22. An integer that divides into another integer
exactly is called a Factor, In other words, if any
integer is divided by another integer without
leaving any remainder, then the latter is the
factor of the former.
For example: 2 is a factor of 8, because 2 divides
evenly into 8.
The quantities that are being multiplied
together to get a product are called factors.
For example: 15 × 4 = 60 implies that 15 and 4
are the factors of 60.

24. Examples:
1. 12x2 – 9x3 = 3x2 (4 – 3x)
2. -10a6b5 – 15a4b6 – 20a3b4
Solution: The common factor of -10, -15 and -20 is -5;
For a6, a4, a3, the common factor is a3
For b5, b6 and b4, the common factor is b4
Therefore, the common monomial factor of the given polynomial is
-5a3b4
After getting the common monomial factor, divide the given polynomial by this
to get the
other factor.
-10a6b5 – 15a4b6 – 20a3b4 = -5a3b4(2a3b + 3ab6 + 4)

25. The difference of two squares is equal to the product of the sum and
difference of the square roots of the terms.
1. 푥2 − 푦2= (x+y) (x-y)
2. 4x2-16y2= (2x+4y)(2x-4y)

26. The square of any binomial is a perfect square trinomial where
the first and the last terms are the square of the
first and square of last term of the binomial and the other term
is a plus (or minus) the product of the first and the
product of the first and last term of the binomial.
x2+ 2xy + y2= (x + y)2
x2 – 2xy + y2= (x – y)2
To check if the trinomial is a perfect square trinomial,
• two terms should be perfect squares
• the other term should be a plus or minus twice the product
of the square roots of the other terms.

27. 1. x3+y3=(x+y)(x2-xy+y2)
2. x3-y3=(x-y)(x2+xy+y2)
3. 27x3 – 8y3 = (3x – 2y) (9x2 + 6xy + 4y2)
4. a3 + 64 = (a + 4)(a2 – 4a + 16)
5. 125b4c – bc4 = bc(125b3 – c3) =
bc(5b – c)(25b2 + 5bc + c2)

28. Sometimes proper grouping of terms is necessary to make the given polynomial
factorable. After terms are
grouped, a complicated expression may be factored easily by applying Types 1 to 5
formulas.
This type of factoring is usually applied to algebraic expressions consisting of at least
four terms.
Examples:
1. 3x(a – b) + 4y(a – b) =(a – b)(3x + 4y)
2. bx + by + 2hx + 2hy = (bx + by) + (2hx + 2hy) grouping
= b(x + y) + 2h(x + y) removal of common
factor from each group
= (x + y)(b+ 2h)
Another solution:
= (bx + 2hx) + (by + 2hy)
= x(b + 2h) + y(b + 2h)
= (b + 2h)(x + y)
3. 3x(2a – b) + 4y(b – 2a) = 3x(2a – b) – 4y(2a – b) alteration of sign
= (2a – b)(3x – 4y)
4. xz – kx + kw – wz = (xz – kx) – (wz – kw)
= x(z – k) – w(z – k)
= (z – k)(x – w)
5. ab3 – 3b2 – 4a + 12 = (ab3 – 3b2) – (4a – 12)
= b2(a – 3) – 4(a – 3)
= (a – 3) (b2 – 4)
= (a – 3) (b – 2) (b+2)

29. A "quadratic" is a polynomial that looks like "ax2
+ bx + c", where "a", "b", and "c" are just
numbers.
For the easy case of factoring, you will find two
numbers that will not only multiply to equal the
constant term "c", but also add up to equal "b",
the coefficient on the x-term. For instance:

30. Factor x2 + 5x + 6.
I need to find factors of 6 that add up to 5. Since 6 can be written as the
product of 2 and 3, and since 2 + 3 = 5, then I'll use 2 and 3. I know from
multiplying polynomials that this quadratic is formed from multiplying two
factors of the form "(x + m)(x + n)", for some numbers m and n. So I'll draw
my parentheses, with an "x" in the front of each:
(x )(x )
Then I'll write in the two numbers that I found above:
(x + 2)(x + 3)
This is the answer: x2 + 5x + 6 = (x + 2)(x + 3)
This is how all of the "easy" quadratics will work: you will find factors of
the constant term that add up to the middle term, and use these factors to
fill in your parentheses.
Note that you can always check your work by multiplying back to get the
original answer. In this case:
(x + 2)(x + 3) = x2 + 5x + 6

31.  Can be written
풂
, b ≠ 0
풃
 Numerator and Denominator
are Polynomials

32. Simplify the following expression:
2푥
푥2
To simplify a numerical fraction, I would cancel off any common
numerical factors. For this rational expression (this polynomial
fraction), I can similarly cancel off any common numerical or
variable factors.
The numerator factors as (2)(x); the denominator factors as (x)(x).
Anything divided by itself is just "1", so I can cross out any factors
common to both the numerator and the denominator. Considering
the factors in this particular fraction, I get:
Then the simplified form of the expression is:
=
2
푥

33. Simplify the following rational expression:
(푥 + 3) (푥 + 4)
(푥 + 3) (푥 + 2)
How nice! This one is already factored for me!
However (warning!), you will usually need to do the
factorization yourself, so make sure you are
comfortable with the process!
The only common factor here is "x + 3", so I'll
cancel that off and get:
Then the simplified form is: