Multiplication of Two or more Trinomials

Multiplication of Two or more Monomials	Multiplication of Two or more Binomials	Multiplication of Two or more Trinomials	Multiplication of Monomial & Binomial	Multiplication of Monomial & Trinomial
Multiplication of Binomial & Trinomial

Before you understand this concept, you are advised to read: What are Trinomials ? What are Like Terms ? What are Terms of Polynomial ? How to add Like Terms ? How to Subtract Like Terms ? Let's understand how to multiply two or more trinomials with the help of following examples: Example 1: Multiply (a + b + c) and (2a + 2b + 2c) Solution: In the given question, we have two trinomials: First trinomial = (a + b + c) Second trinomial = (2a + 2b + 2c) Write in the multiplication expression and we get: (a + b + c) (2a + 2b + 2c) Use Distributive Law and multiply each term of first trinomial with every term of second trinomial & this is done in the following steps: = a(2a + 2b + 2c) + b(2a + 2b + 2c) + c(2a + 2b + 2c) = (a X 2a) + (a X 2b) + (a X 2c) + (b X 2a) + (b X 2b) + (b X 2c) + (c X 2a) + (c X 2b) + (c X 2c) = 2a2 + 2ab + 2ac + 2ab + 2b2 + 2bc + 2ac + 2bc + 2c2 Rearrange the terms and we get: = 2a2 + 2c2 + 2b2 + 2ab + 2ab + 2bc + 2bc + 2ac + 2ac Add like terms and we get: = 2a2 + 2c2 + 2b2 + 4ab + 4bc + 4ac Hence, (a + b + c) (2a + 2b + 2c) = 2a2 + 2c2 + 2b2 + 4ab + 4bc + 4ac Example 2: Multiply (x2 + 2x + 1) and (2x2 + x + 2) Solution: In the given question, we have two trinomials: First trinomial = (x2 + 2x + 1) Second trinomial = (2x2 + x + 2) Write in the multiplication expression and we get: (x2+ 2x + 1) (2x2 + x + 2) Use Distributive Law and multiply each term of first trinomial with every term of second trinomial & this is done in the following steps: = x2(2x2 + x + 2) + 2x(2x2 + x + 2) + 1(2x2 + x + 2) = (x2 X 2x2) + (x2 X x) + (x2 X 2) + (2x X 2x2) + (2x X x) + (2x X 2) + (1 X 2x2) + (1 X x) + (1 X 2) = 2x4 + x3 + 2x2 + 4x3 + 2x2 + 4x +2x2 + x + 2 Rearrange the term as shown below: = 2x4 + x3 + 4x3 + 2x2 + 2x2 + 2x2 + 4x + x + 2 Add like terms and we get: = 2x4 + 5x3 + 6x2 + 5x + 2 Hence, (x2+ 2x + 1) (2x2 + x + 2) = 2x4 + 5x3 + 6x2 + 5x + 2 Example 3: Multiply (a + b + 1 ), (a + b + 3 ) and (2a + 3b + 2) Solution: In the given question, we have three trinomials: First trinomial = (a + b + 1) Second trinomial = (a + b + 3) Third trinomial = (2a + 3b + 2) Write in the multiplication expression and we get: (a + b + 1) (a + b + 3) (2a + 3b + 2) Use Distributive Law and multiply each term of first trinomial with every term of second trinomial and keep the third trinomial as such & this is done in the following steps: = [ (a(a + b + 3) + b(a + b + 3) + 1(a + b + 3) ] (2a + 3b + 2) = [(a X a) + (a X b) + (a X 3) + (b X a) + (b X b) + (b X 3) + (1 X a) + (1 X b) + (1 X 3)] (2a + 3b + 2) = [ a2 + ab + 3a + ab + b2 + 3b + a + b + 3] (2a + 3b + 2) Rearrange the term as shown below: = [ a2 + b2 + 3a + a + 3b + b + ab + ab + 3] (2a + 3b + 2) Add like terms and we get: =[a2 + b2 + 4a + 4b + 2ab + 3] (2a + 3b + 2) Now again use Distributive Law and multiply each term of polynomial [a2 + b2 + 4a + 4b + 2ab + 3] with every term of third trinomial i.e. (2a + 3b + 2) & this is done in the following steps: = a2(2a + 3b + 2) + b2(2a + 3b + 2) + 4a(2a + 3b + 2) + 4b(2a + 3b + 2) + 2ab(2a + 3b + 2) + 3(2a + 3b + 2) = (a2 X 2a) + (a2 X 3b) + (a2 X 2) + (b2 X 2a) + (b2 X 3b) + (b2 X 2) + (4a X 2a) + (4a X 3b) + (4a X 2) + (4b X 2a) + (4b X 3b) + (4b X 2) + (2ab X 2a) + (2ab X 3b) + (2ab X 2) + (3 X 2a) + (3 X 3b) + (3 X 2) = 2a3 + 3a2b + 2a2 + 2ab2 + 3b3 + 2b2 + 8a2 + 12ab + 8a + 8ab + 12b2 + 8b + 4a2b + 6ab2 + 4ab + 6a + 9b + 6 Rearrange the terms as shown below: = 2a3 + 3b3 + 2a2 + 8a2 + 2b2 + 12b2 + 3a2b + 4a2b + 2ab2 + 6ab2 + 8a + 6a + 8b + 9b + 4ab + 12ab + 8ab + 6 Add Like terms and we get: = 2a3 + 3b3 + 10a2 + 14b2 + 7a2b + 8ab2 + 14a + 17b + 24ab + 6 Hence, (a + b + 1) (a + b + 3) (2a + 3b + 2) = 2a3 + 3b3 + 10a2 + 14b2 + 7a2b + 8ab2 + 14a + 17b + 24ab + 6