Relations and Functions

The Four Rules / Operations of Mathematics which includes Addition, Subtraction, Multiplication and Division can be used to Simplify Algebraic Expressions.
Before making use of the Four Operations of Algebraic Expressions, Let's know Signed Numbers
It is the Basis of Signed Numbers that helps us solve Algebraic Expressions.
Signed Numbers are Negative or Positive Numbers that are expressed in Algebraic Expression.
Ex: -5 , 3, -11 , -2, +19 etc.
Note that any number without a sign, the default sign is Positive (+).
let's say, if I Put Four numbers together like this 6 - 5 - 8 + 2
and I ask to give the Sign of Each numbers
The sign of 6 is Positive (+)
The sign of -5 is Negative (-)
The sign of -8 is Negative (-)
The sign of +2 is Positive (+)
Remember always that the sign of the number is always called first before the Number,and any number without a Sign, the default sign is Positive (+).
Getting these in Mind, Let's observe the sign Rules.

Addition and Subtraction Rules
+ + + = Add the two Numbers and end up in Positive (+)
- + - = Add the two Numbers and end up in Negative (-)
+ + - = Subtract the smaller number from the bigger number and carry the Sign of the Bigger Number
- + + = Subtract the smaller number from the bigger number and carry the Sign of the Bigger Number
Short cut Rule to this
Same Sign, add them and carry that sign.
Opposite Sign, Subtract the smaller number from the Bigger number and Carry the Sign of the Bigger Number.
Example : 1. Simplify 6 + 5
Ans : the 6 is Positive and the 5 is Positive, Therefore, they are same Sign, we add them, this gives us 11. the answer remains like this because, any number without a sign, the sign is Positive.
Example : 2. Simplify -3 - 12
Ans : the -3 is Negative and the -12 is Negative, Therefore, they are same Sign, we add them, this gives us 15 ,because they are negative, the answer becomes -15
Example : 3. Simplify -9 - 5
Ans : the -9 is Negative and the -5 is Negative, Therefore, they are same Sign, we add them, this gives us 14 ,because they are negative, the answer becomes -14
Example : 4. Simplify -3 + 8
Ans : the -3 is Negative and the +8 is Positive, Therefore, they have different Signs, we subtract 3 from 8 , this gives us 5 ,because the bigger number here is 8 and it is Positive, the answer becomes 5 (which is Positive 5)
Example : 5. Simplify 3 - 8
Ans : the 3 is Positive and the -8 is Negative, Therefore, they have different Signs, we subtract 3 from 8 , this gives us 5 ,because the bigger number here is 8 and it is Negative, the answer becomes -5

Note: These Sign Rules will be used Through out your Mathematical Journey until we all depart this world. Practice using your own examples and the examples we provided for you.

We can only add or subtract like terms to give a single term. This is often called Collecting Like Terms..
< Remember > : When you notice like terms, add or subtract the Coefficient and attach the Variables with their respective Exponents.
It is easier to simplify, if you rewrite the expressions with like terms next to each other.
Example 1: Simplify 13x⁴ - 7y²- 18x⁴ + 10y²
Example 2: Simplify -12x⁴y² + 8y²- 18x⁴ + 10y²
Example 3: Simplify 5x + 4 - 9y + 3x + 2y - 7
Example 4: Simplify 4p²y + 5py² + 3p²y - 10py²
Example 5: Simplify x² + x + 2x²

Multiplication Sign Rules
+ x + = +
- x - = +
+ x - = -
- x + = -
Short Cut to the Rules
When you multiply same signs, the result is Positive
When you multiply opposite Signs, the result is Negative

Note: These Sign Rules will be used Through out your Mathematical Journey until we all depart this world. Practice using your own examples and the examples we provided for you.

When you are multiplying Algebraic Expressions, First make use of the Multiplication Sign Rules and next, make use of the Rules of Indices.
Rule of Indices
When you multiplying same Variables , add the Exponents.
Only multiply the coefficients .
Example: P⁴ x P⁷ = P¹¹
Note that when there is no Exponent shown, the default Exponent is 1.
Note also that we use Bracket to show multiplication.
Example 1: simplify 4P x 8P⁴
Example 2: simplify 2st ³ x 4s⁴t²
Example 3: simplify 5k⁵ r ³ ( 4k⁴r²)
Example 4: -2a x 4c x 5b
Example 5: 12ab(-5ax)
Example 6: (-3a⁵b³)(-7a³b⁴)

Brackets are used to group terms together and may also be used for Multiplication. An expression with Brackets in it can be replaced by an equivalent one without brackets. This is called Removing the Brackets or Expanding the Expression.
To remove the Bracket, we use the Distributive law. Multiply the Term outside the bracket by each of the terms inside the Bracket.
Example 1: Simplify 4a(2a + 3)
Example 1: Simplify 4(x + 1) + 2(2x + 1)
Example 1: Simplify -3(2x - 4) + 6(2x - 5)
Example 1: Simplify 2a[(a + 3b) + 4(2a - b)]

A Binomial is an expression of two terms separated by a Plus (+) or Minus (-) signs.
For example: (a + 3), (x ²- y⁵), (2x + 3y) and (3x - 2) are all Binomial Expressions .
The multiplication of two binomials requires the use of the Distributive Property.
Multiply each term in one Bracket by each term in the Other Brackets. We also consider this as the FOIL Method.
F = First Term
O = Outer Term
I = Inner Term
L = Last Term
This means that the First Term will multiply by the Inner Term and Last Term
The Outer Term will also be multiply by the Inner Term and Last Term.
Example : in (a + 2)(b - 3) , a is the First Term, 2 is the Outer Term, b is the Inner Term, -3 is the Last Term.
To Expand (a + 2)(b - 3) Example 2: (x - 2)(x + 3)
Example 2: (y + 2)(x - 4)
Example 2: (x - 7)²
Example Expand and Simplify 2: (2x + y)(x - y) + (2x - y)(x + y)
Example Expand and Simplify 2: (4n + 2m)(2n + 3m) - (n - m)(3n + 2m)
Example 2: (2x - 3)³
Example 2: (3y + 4z)³
Example 2: (x - 2y)⁴

Write Exercise Here

Write Self Prace Here.

Write Exercise Answer Here