**What is a trinomial?**

A trinomial is a polynomial that consists of the indicated sum of three different terms, that is, it is built by algebraically adding three monomials of different degrees, either of one or more variables. They are very common polynomials in algebra.

Some examples of trinomials are the following:

x2 + 5x – 3 (trinomial of degree 2) x– x2 – 6×3 (trinomial of degree 3) –7xy2 + 4x2y – x3 (trinomial of absolute degree 3, of degree 3 in x and of degree 2 in y)

The first and second of these trinomials are of a single variable, in this case the variable “x”, while the third trinomial is of two variables “x” and “y”.

**Examples of trinomials**

There are several types of trinomials that occur in numerous applications, among which are:

**perfect square trinomial**

A perfect square trinomial is obtained by expanding the square of a sum or the square of a difference of terms. Both developments are known by the name of *remarkable products*.

First, we have the square of the sum: (a + b)2. Expanding this expression, we get:

(a + b)2 = (a + b) × (a + b) = a2 + a∙b + b∙a + b2

The two middle terms are identical and reduce to 2a∙b, therefore:

(a + b)2 = a2 + 2a∙b + b2

The trinomial a2 + 2a∙b + b2 contains two perfect squares: a2 and b2, while the remaining term is equal to twice the product of the two terms of the original binomial.

The square of a difference results in a trinomial similar to the previous one, except for a negative sign that affects the double product of the terms of the original binomial:

(a − b)2 = (a − b) × (a − b) = a2 − a∙b − b∙a + b2

Once again, the like terms are reduced to a single term and it is obtained that:

(a − b)2 = a2 − 2a∙b + b2

It is no longer possible to further reduce the result.

These remarkable products, easily memorized, associate a perfect square trinomial with the square of the corresponding binomial, for example:

(x − 5)2 = x2 − 10∙x + 25 (2y + 3)2 = 4y2 + 12∙y + 9

It should be noted that not all perfect square trinomials have one variable or degree 2. Here are examples of this type of trinomials with two and more variables and also with degrees different from 2:

(x + y)2 = x2 + 2∙xy + y2 (2z2 + y)2 = 4z4 + 4∙z2y + y2 (5xy3 − z)2 = 25x2y6 − 10 xy3z + z2

**Trinomial of the form x2 + bx + c**

In this trinomial only one of the terms is a perfect square, in this case it is x2 and its numerical coefficient is 1. The next term b⋅x is linear and the last term is the independent term. Examples of this class of trinomials are:

x2 + 5∙x + 6 (b = 5; c = 6) y2 − 4∙y + 3 (b = −4; c = 3) m2 − 12∙m + 11 (b = −12; c = 11)

**Trinomial of the form ax2 + bx + c**

It is similar to the previous ones, except that the coefficient of the quadratic term is different from 1, as in these trinomials:

3×2 − 5∙x − 2 (a = 3; b = −5; c = −2) 6y2 + 7∙y + 2 (a=6; b = 7; c = 2) 2m2 + 29∙m + 90 ( a=2; b=29; c=90)

**Factoring Trinomials**

A very frequent algebraic operation is the factorization of trinomials, which consists of writing them as the product of factors different from 1. There are specific procedures for each of the trinomials described.

**Factoring Perfect Square Trinomials**

They can be factored by inspection from the notable products:

(a + b)2 = a2 + 2a∙b + b2

(a − b)2 = a2 − 2a∙b + b2

The steps to factor a perfect square trinomial are:

1.- Verify that the trinomial contains two perfect squares a2 and b2, both terms must be preceded by the same sign, usually the + sign. If both are preceded by a − sign, this can be factored without problem.

2.- Determine the values of a and b by extracting the square root of a2 and b2.

3.- Check that the third term is the double product of a and b.

**Factorization of trinomials of the form x2 + bx + c**

This is the trinomial with a unique quadratic term, to factor it, it is written as the product of two binomials:

x2 + bx + c = (x + r) ∙ (x + s)

where r and s are two numbers to determine.

Note that by developing the right hand side, using the distributive property, we obtain:

(x + r) ∙ (x + s) = x2 + s∙x + r∙x + r∙s = x2 + (r+s)∙x + r∙s

So, for this expression to reflect the original trinomial, the numbers u and v must meet the following conditions:

r∙s = c

r + s = b

Some trinomials of the form x2 + bx + c do not admit factorization by this method, however they can be factored with the help of the general formula or solvent formula.

**Factorization of trinomials of the form ax2 + bx + c**

A procedure to factor this type of trinomials is:

Multiply and divide the trinomial by the coefficient “a” Perform the product between “a” and the first and third terms of the trinomial, leaving the product with the second term without performing. The procedure described in the previous section is applied to the trinomial obtained, that is, it is written as the product of two binomials, but in this case the first term of each binomial is not “x”, but “a∙x”. We look for two numbers r and s such that a∙c = r∙s and also r + s = b Finally, the resulting binomials are simplified as much as possible, see solved exercise 3.

**solved exercises**

**Exercise 1**

Find the trinomial that results from developing the following notable product: (4x − 3y)2

The remarkable product formula for the square of a difference is applied, giving as a result:

(4x − 3y)2 = (4x)2 − 2∙4x∙3y + (3y)2 = 16×2 − 24∙xy + 9y2

**Exercise 2**

Factor the following trinomial:

x2 + 5x + 6

This is a trinomial of the form x2 + bx + c, with b = 5 and c = 6, so you can try to factor it with the procedure described above. To do this, we must find two numbers r and s such that multiplied to obtain 6 and added to 5:

r∙s = 6 and r + s = 5.

The numbers sought are r = 3 and s = 2, since they meet these conditions, therefore:

x2 + 5x + 6 = (x + 3)(x + 2)

It is left as an exercise for the reader to verify that by developing the right hand side the original trinomial is easily reached.

**Exercise 3**

Factor 3×2 − 5x − 2.

This is a trinomial of the form ax2 + bx + c, with a = 3, b = −5, and c = −2. The process is:

-Multiply and divide by a =3:

Make the product of «a» by the first and the third term, indicating the product with the second term:

Now we have to write the product of two binomials, whose first term is 3x and find two numbers r and s such that:

When multiplied, they give −6 And when added algebraically, they give −5

These numbers are r = −6 and s = 1:

Finally, the resulting product of binomials is simplified:

**proposed exercises**

Factor the following trinomials:²

x² – 14x + 49 p² + 12pq + 36q² 12x² – x – 6 z² + 6z + 8

**References**

Baldor. 1977. Elementary Algebra. Venezuelan Cultural Editions. Jiménez, R. 2008. Algebra. Prentice Hall. Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning. Zill, D. 1984. Algebra and Trigonometry. 1st Edition. McGraw Hill. Zill, D. 2008. Precalculus with calculus advances. 4th. Edition. McGraw Hill.