4. Types of algebraic expressions. Trinomials

Like terms : The terms having the same literal factors are called like or similar terms.
In the algebraic expression 12a² – 15b² + b² –17a² + 8ab + 9, we have, 12a² and –17a² as like terms and also – 15b² and b² are like terms. 

Unlike terms :  The terms not having same literal factors are called unlike or dissimilar terms.

In the algebraic expression 3p²q + 5pq² – 7pq – 9qp², 5pq² and –7pq are unlike terms.

Types of algebraic expressions. Trinomials

A trinomial algebraic expression is an algebraic expression that has three terms.
The prefix 'tri' in 'trinomial' stands for 3 terms.
To add two trinomials, you must:
 1) remove the brackets (without changing the signs, because the “+” sign is in front of the brackets);
2) add the like terms.

Example:
Add trinomials: 

(−5x³+3y−5y²)+(8x³+5y²−2y)

1) Remove the brackets:
(−5x³+3y−5y²)+(8x³+5y²−2y)==−5x³+3y−5y²+8x³+5y²−2y.

2) Find the like terms and add:
−5x³¯¯¯¯¯¯¯¯+3y−5y²+8x³¯¯¯¯¯¯¯¯+5y²−2y=3x³+3y¯¯¯¯¯¯−5y²+5y²−2y¯¯¯¯¯¯=3x³+y.
To multiply a trinomial by a trinomial, you need to multiply each term of one trinomial by each terms of another trinomial and add the resulting products. 
(a+b+c)⋅ (d+e+f)==a⋅d + a⋅e +a⋅f +b⋅d +b⋅e +b⋅f +c⋅d +c⋅e +c⋅f = ad + ae + af + bd + be + bf + cd + ce + cf

To subtract two trinomials , you must:

1) remove the brackets and change the signs of the trinomials that are preceded by the sign "−" to the opposite;

2) combine the like terms of the trinomials.

Example:

Let us calculate the difference of trinomials (7x²+3x−2) and −2x²+2x+3

1) Write down the difference of the trinomials and remove the brackets, taking the signs before the brackets into account:
(7x²+3x−2)−(−2x²+2x+3)=7x²+3x−2+2x²−2x−3

2) Find the like terms:
7x²¯¯¯¯¯+3x¯¯¯¯¯¯¯¯−2 +2x²¯¯¯¯¯−2x¯¯¯¯¯¯¯¯−3

3) Combine the like terms:
7x²¯¯¯¯¯+3x¯¯¯¯¯¯¯¯¯¯¯¯−2+2x²¯¯¯¯¯¯¯¯−2x¯¯¯¯¯¯¯¯¯¯¯¯−3=(7+2)x²+(3−2)x−2−3=9x²+1x−5

4) If the coefficient of a term is 1, then usually we do not write it:
9x²+1x−5=9x²+x−5