In algebraic expression consisting of only one term is called a monomial, two terms is called a binomial and when there are three terms separated by an addition or a subtraction operation is called a trinomial. ‘Tri’ in the word trinomial means three and hence the name. It is also referred to as a polynomial as ‘poly’ means more than two terms. For example, a+3b-c consists of three terms a, b and c and hence a trinomial. In 3b, 3 is called the coefficient of b. In a given polynomial we come across like terms and unlike terms.
Like terms are the terms which have the same variable or literal but a different coefficient. For example, (3b, -7b);(5x2, 12x2);(-4xy, 7xy) are some of the like terms. Unlike terms as the name suggests are the terms which have different variables. For example, (7x, 8y); (4ab, -4ac); (2y2, -2x2) etc are some of the unlike terms.
It is not necessary that there would be only two like or unlike terms, it depends on the number of terms in the given polynomial. Now let us learn as to How do you multiply trinomials. There are two methods in which we can multiply trinomials, one is the horizontal method the other is the vertical method.
The steps to be followed in a Horizontal method of multiplication are as follows:
Example: Multiply (4x2-3x+5)(x2+5x-3)
Solution: First and second terms are chosen irrespective of the order
Let (4x2-3x+5) be the first term and (x2+5x-3) be the second term
Arrange the two polynomials horizontally
(4x2-3x+5) X (x2+5x-3)
Now distribute each of the terms of the first trinomial with each of the terms of the second trinomial
=4x2(x2+5x-3)- 3x(x2+5x-3) +5(x2+5x-3)
=4x4+20x3-12x2 - 3x3-15x2+9x +5x2+25x-15
Combine the like terms and simplify
=4x4+x3(20-3)+x2(-12-15+5)+ x(9+25) – 15
= 4x4+17x3-22x2+34x-15
The steps to be followed in a Vertical method of multiplication are as follows:
Example: Multiply (4x2-3x+5)(x2+5x-3)
The first and the second terms are chosen irrespective of the order
Let (4x2-3x+5) be the first term and (x2+5x-3) be the second term
Arrange the two polynomials in a vertical form
4x2-3x+5
X x2+5x-3
Working from right towards left each term of the lower trinomial is multiplied with each of the terms of the upper trinomials. Then the products are written underneath the second trinomial in three rows in the order of the degree of each term and then the like terms simplified as shown below
4x2-3x+5
X x2+5x-3
-12x2+ 9x – 15
+20x3-15x2 + 25x
4x4 - 3x3 + 5x2 .
4x4+17x3-22x2+ 34x-15 is the final product!
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