In algebra, we study about equations and functions. Polynomials are very useful topic which is used very often allover the mathematics. Polynomials are said to the expressions usually having many terms. The general form of a polynomial is -

P (x) = $b_{n}x^{n}+b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+...+b_{0}$

On the basis of number of terms, we may following three types -**(1)** Monomial - Polynomial with one term.**(2)** Binomial** **- Polynomial with two terms.**(3)** Trinomial - Polynomial with three terms.

In this page below, we shall learn about trinomials. Trinomials are polynomials containing total number of three terms. For example :

Trinomials can be applied various operations just as other polynomials, like - addition, subtraction, multiplication and division. Especially, we are going to study about multiplication of trinomials. The distributive method can be used to multiply two trinomials. In this case, multiplicand and the multiplier both are trinomials. Multiplication of the trinomials can be done by either the horizontal method or the vertical method of multiplication. Let us go ahead and learn how to multiply two or more trinomials together.

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To multiply two trinomials, we will have to multiply each term of the second trinomial by the first term of the first trinomial and then repeat the multiplication by multiplying each term of the second trinomial by the second term of the first trinomial and finally, multiply each term of the second trinomial by the third term of the first trinomial. This can be done by either the horizontal method or the vertical method of multiplication. Now, group the like terms together and add them.

Given below are some of the examples in solving trinomials multiplication.

(2x^{2} + 2x + 1) ----- First trinomial

(3x^{2} + 5x + 2) ----- Second trinomial

Let us use horizontal method of multiplication.

**Step 1:**

Multiply the first term of the first trinomial with each term of second trinomial.

2x^{2} × (3x^{2} + 5x + 2) = 6x^{4} + 10x^{3} + 4x^{2} ....................(1)

Next, multiply the second term of the first trinomial with each term of the second trinomial.

2x × (3x^{2} + 5x + 2) = 6x^{3} + 10x^{2} + 4x ....................(2)

Finally, multiply the third term of the first trinomial with each term of the second trinomial.

1 × (3x^{2} + 5x + 2) = 3x^{2} + 5x + 2 ...................(3)

**Step 2:**

Add (1), (2) and (3)

6x^{4} + 10x^{3} + 4x^{2} + 6x^{3} + 10x^{2} + 4x + 3x^{2} + 5x + 2

Group the like terms together.

6x^{4} + 10x^{3} + 6x^{3} + 4x^{2} + 10x^{2} + 3x^{2} + 4x + 5x + 2

Now, add the like terms and simplify.

6x^{4} + 16 x^{3} + 17x^{2} + 9x + 2

Hence, the product of (2x^{2} + 2x + 1) and (3x^{2} + 5x + 2) is 6x^{4} + 16 x^{3} + 17x^{2} + 9x + 2.

Multiplying binomials and trinomials is same as we multiply trinomials. Let us see with the help of examples how binomials are multiplied by trinomials:

(x + 3) (Binomial)

(x^{2} + y + 5) (Trinomial)

Multiply each term of the binomial by the each term of the trinomial.

(x + 3)(x^{2} + y + 5) = x(x^{2} + y + 5) + 3(x^{2} + y + 5)

= (x * x^{2}+ x * y + 5x) + 3x^{2}+ 3y + 3 * 5

= x^{3}+ xy + 5x + 3x^{2} + 3y + 15

= x^{3} + 3x^{2 }+ xy + 5x + 3y + 15

Therefore, the product of (x + 3) and (x^{2} + y + 5) is x^{3} + 3x^{2 }+ xy + 5x + 3y + 15.

(x + y - 3) ..........Trinomial

(y + 5)................Binomial

(x + y - 3)(y + 5) = x(y + 5) + y(y + 5) - 3(y + 5)

= xy + 5x + y^{2} + 5y - 3y - 15

= xy + 5x + y^{2} + 2y - 15 (Combine like terms)

Therefore, the product of (x + y - 3) and (y + 5) is xy + 5x + y^{2} + 2y - 15.

Given: (3x + 2y + 6)(4x + y + 3) = 0

Multiplying the terms

$\rightarrow$ 12 x^{2} + 3xy + 9x + 2y^{2} + 8xy + 6y + 24x + 6y + 18 = 0

Combine like terms

$\rightarrow$ 12x^{2} + 2y^{2} + 11xy + 33x + 12y + 18 = 0

So, by multiplication, we will get second degree and third degree polynomials

(x^{2} + 2x + 4) ----- First Trinomial

(x^{2} + 6x + 5) ----- Second Trinomial

Let us follow** vertical method **of multiplication.

Hence, the product of (x^{2} + 2x + 4) and (x^{2} + 6x + 5) is x^{4} + 8x^{3} + 21x^{2} + 34x + 20.

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