Trinomials
A trinomial is an algebraic expression that has three terms. An algebraic expression consists of variables and constants of one or more terms. These expressions use symbols or operations as separators such as +, –, ×, and ÷. A trinomial along with monomial, binomial, and polynomial are categorized under this algebraic expression. Let us learn more about trinomials, factoring trinomials, the formula for factoring trinomials along with solving a few examples.
1.  What is a Trinomial? 
2.  Perfect Square Trinomial 
3.  Quadratic Trinomial 
4.  How to Factor Trinomials? 
5.  Factoring Trinomials Formula 
6.  FAQs on Trinomials 
What is a Trinomial?
A trinomial is an algebraic expression that has three nonzero terms and has more than one variable in the expression. A trinomial is a type of polynomial but with three terms. A polynomial is an algebraic expression that has one or more terms and is written as \[a_0x^n + a_1x^{n1} + a_2x^{n2}...........+ a_nx^0\] in the standard form. Where \(a_0, a_1, a_2.........a_n\) are constants and n is a natural number. Whereas a trinomial can be expressed with multiple variables and three terms. For examples: x^{2} + y^{2} + xy, 5x^{2}  4x^{2} + z and xyz^{3} + x^{2}z^{2} + zy^{3}. Examples of trinomials with one variable are: x^{2} + 2x + 3, 5x^{4}  4x^{2} +1 and \(7y  \sqrt{3}  y^2\).
A polynomial can be referred by different names depending on the number of terms it has. The table below mentions the names.
Number of terms  Polynomial  Example 
1  Monomial  xy 
2  Binomial  x + y 
3  Trinomial  x^{2} + xz + 1 
Perfect Square Trinomial
A perfect square trinomial is defined as an algebraic expression that is obtained by squaring a binomial expression. It is of the form ax^{2} + bx + c. Here a, b, and c are real numbers and a ≠ 0. For example, let us take a binomial (x + 2) and multiply it to (x + 2). The result obtained is x^{2} + 4x + 4. A perfect square trinomial can be decomposed into two binomials and the binomials when multiplied with each other gives the perfect square trinomial.
Quadratic Trinomial
A quadratic trinomial is a type of algebraic expression with variables and constants. It is expressed in the form of ax^{2} + bx + c, where x is the variable and a, b, and c are nonzero constants and integers. The constant 'a' is known as a leading coefficient, 'b' is the linear coefficient, 'c' is the additive constant. A quadratic trinomial also describes the discriminant D where it defines the quantity of an expression and it is written as D = b^{2}  4ac. The discriminant helps in classifying among the different cases of quadratic trinomials. If the value of a quadratic trinomial with a single variable is zero, then it is known as a quadratic equation i.e ax^{2} + bx + c = 0.
How to Factor Trinomials?
Factoring a trinomial means expanding an equation into the product of two or more binomials. It is written as (x + m) (x + n). A trinomial can be factorized in many ways. Let's discuss each case.
Quadratic Trinomial in One Variable
The general form of quadratic trinomial formula in one variable is ax^{2} + bx + c, where a, b, c are constant terms and neither a, b, or c is zero. For the value of a, b, c, if b^{2}  4ac > 0, then we can always factorize a quadratic trinomial. It means that ax^{2} + bx + c = a(x + h)(x + k), where h and k are real numbers. Now let's learn how to factorize a quadratic trinomial with an example.
Example: Factorize: 3x^{2}  4x  4
Solution:
Step 1: First multiply the coefficient of x^{2} and the constant term.
3 × 4 = 12
Step 2: Break the middle term 4x such that on multiplying the resulting numbers, we get the result 12 (obtained from the first step).
4x = 6x + 2x
6 × 2 = 12
Step 3: Rewrite the main equation by applying the change in the middle term.
3x^{2}  4x  4 = 3x^{2}  6x + 2x  4
Step 4: Combine the first two terms and the last two terms, simplify the equation and take out any common numbers or expressions.
3x^{2}  6x + 2x  4 = 3x (x  2) + 2(x  2)
Step 5: Again take (x  2) common from both the terms.
3x (x  2) + 2(x  2) = (x  2) (3x + 2)
Therefore, (x  2) and (3x + 2) are the factors of 3x^{2}  4x  4.
Quadratic Trinomial in Two Variable
There is no specific way to solve a quadratic trinomial in two variables. Let's take an example.
Example: Factorize: x^{2} + 3xy + 2y^{2}
Solution
Step 1: These types of trinomials also follow the same rule as above, i.e., we need to break the middle term.
x^{2} + 3xy + 2y^{2 }= x^{2} + 2xy + xy + 2y^{2}
Step 2: Simplify the equation and take out common numbers of expressions.
x^{2} + 2xy + xy + 2y^{2 }= x (x + 2y) + y (x + 2y)
Step 3: Again take (x + 2y) common from both the terms.
x (x + 2y) + y (x + 2y) = (x + y) (x + 2y)
Therefore, (x + y) and (x + 2y) are the factors of x^{2} + 3xy + 2y^{2}
If Trinomial is an Identity
Let's see some algebraic identities that are mentioned in the table below:
Identity  Expanded Form 
(x + y)^{2}  x^{2} + 2xy + y^{2} 
(x  y)^{2}  x^{2}  2xy + y^{2} 
(x^{2}  y^{2})  (x + y) (x  y) 
Example: Factorize: 9x^{2} + 12xy + 4y^{2}
Solution:
Step 1: Identify which identity can be applied in the expression.
We can apply (x + y)^{2} = x^{2} + 2xy + y^{2}
Step 2: Rearrange the expression so that it can appear in the form of the above identity.
9x^{2} + 12xy + 4y^{2 }= (3x)^{2} + 2 × 3x × 2y + (2y)^{2}
Step 3: Once the expression is arranged in the form of the identity, write its factors.
(3x)^{2} + 2 × 3x × 2y + (2y)^{2 }= (3x + 2y)^{2} = (3x + 2y) (3x + 2y)
Therefore, (3x + 2y) is the factor of 9x^{2} + 12xy + 4y^{2}.
Leading coefficient of 1
Let us look at an example.
Example: Factorize x^{2 }+ 7x + 12
Solution:
Step 1: Compare the given equation with the standard form to obtain the coefficients.
ax^{2} + bx + c is the standard form, comparing the equation x^{2} + 7x + 12 we get a = 1, b = 7, and c = 12
Step 2: Find the paired factors of c i.e 12 such that their sum is equal to b i.e 7.
The pair factor of 12 are (1, 12), (2, 6), and (3, 4). Therefore, the suitable pair is 3 and 4.
Step 3: Add each number to x separately.
(x + 3) (x + 4)
Therefore, (x + 3) (x + 4) are the factors for x^{2 }+ 7x + 12.
Factorizing with GCF
When the trinomial needs to be factorized where the leading coefficient is not equal to 1, the concept of GCF(Greatest Common Factor) is applied. Let us see the steps:
 Write the trinomial in descending order, from highest to lowest power.
 Find the GCF by factorization.
 Find the product of the leading coefficient 'a' and the constant 'c.'
 Find the factors of the product 'a' and 'c'. Pick a pair that sums up to get the number instead of 'b'.
 Rewrite the original equation by replacing the term “bx” with the chosen factors.
 Factor the equation by grouping.
Factoring Trinomials Formula
A trinomial can be a perfect square or a nonperfect square. We have two formulas to factorize a perfect square trinomial. But for factorizing a nonperfect square trinomial, we do not have any specific formula, instead, we have a process.
 The factoring trinomials formulas of perfect square trinomials are:
a^{2} + 2ab + b^{2} = (a + b)^{2}
a^{2}  2ab + b^{2} = (a  b)^{2}
For applying either of these formulas, the trinomial should be one of the forms a^{2} + 2ab + b^{2} (or) a^{2}  2ab + b^{2}.
 The process of factoring a nonperfect trinomial ax^{2} + bx + c is:
Step 1: Find ac and identify b.
Step 2: Find two numbers whose product is ac and whose sum is b.
Step 3: Split the middle term as the sum of two terms using the numbers from step  2.
Step 4: Factor by grouping.
To factorize a trinomial of the form ax^{2} + bx + c, we can use any of the belowmentioned formulas:
 a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b)
 a^{2}  2ab + b^{2} = (a  b)^{2} = (a  b) (a  b)
 a^{2}  b^{2} = (a + b) (a  b)
 a^{3} + b^{3} = (a + b) (a^{2}  ab + b^{2})
 a^{3}  b^{3} = (a  b) (a^{2} + ab + b^{2})
Related Topics
Here is a list of topics that are related to trinomials.
Examples on Trinomials

Example 1: Help Tim find the factors of x^{2}  5x + 6.
Solution:
x^{2}  5x + 6
= x^{2}  3x  2x + 3 × 2
= x(x  3)  2x + 6
= x(x  3)  2(x  3)
= (x  2)(x  3)
Therefore, (x  2)(x  3) is the factors of x^{2}  5x + 6.

Example 2: Help Clara find the factors of 15a^{2} + 38ab + 24b^{2}.
Solution:
Here we need to break the middle term.
First, multiply the coefficient of a^{2} and b^{2} = 15 × 24 = 360.
Now find two numbers such that on multiplication they give the result 360 and on addition, they give the result 38.
18 × 20 = 360 and 18 + 20 = 38
15a^{2} + 38ab + 24b^{2}
= 15a^{2} + 18ab + 20ab + 24b^{2}
= 3a(5a + 6b) + 4b(5a + 6b)
= (5a + 6b)(3a + 4b)
Therefore, (5a + 6b)(3a + 4b) is the factors for 15a^{2} + 38ab + 24b^{2}.

Example 3: If y  3 is a factor of y^{2} + a  6y, then find the value of a. Find the other factor of the trinomial.
Solution:
(y  3) is a factor of y^{2} + a  6y. Then if we put (y = 3) in the trinomial y^{2} + a  6y, its value will be 0.
3^{2} + a  6 × 3 = 0
9 + a  18 = 0
a  9 = 0
a = 9
Now factorize the trinomial y^{2} + a  6y = y^{2} + 9  6y
The above trinomial is the expansion of the identity (x  y)^{2} = x^{2}  2xy + y^{2}
y^{2}  6y + 9
= y^{2 } 2 × 3 × y + 3^{2} = (y  3)^{2}
Therefore, a = 9 and y^{2} + a  6y = (y  3)^{2}.
FAQs on Trinomials
What is a Trinomial?
A trinomial is an algebraic expression that has three nonzero terms and has more than one variable in the expression. For example: x^{2} + 5y  25, a^{3}  16b + 10. These are trinomials as they have three terms i.e. coefficient, variables, and constants. A trinomial can have only one variable or two variables.
What is a Perfect Square Trinomial?
Perfect square trinomials are the trinomials that follow the identity (a + b)^{2 }= a^{2} + 2ab +b^{2}. It is an algebraic expression that is obtained by squaring a binomial expression. It is of the form ax^{2} + bx + c. Here a, b, and c are real numbers, and a ≠ 0. For example, x^{2} + 2x + 1 and 4y^{2} −20y + 25.
How Do You Factor a Trinomial?
A trinomial can be factored in the form x^{2} + bx + c. First, we need to find two integers (y and z) whose product sums up to c and through addition, it sums up to b. Once we find the two numbers, we rewrite the trinomial as x^{2} + yx + zx + c and use the grouping and distributive property of the factor to find out the factors of the expression. The factors will be (x + y) (x + z).
What is a Quadratic Trinomial?
A quadratic trinomial is a polynomial with three terms and the degree of the trinomial must be 2. It means that the highest power of the variable cannot be greater than 2. For example: x^{2} + y^{2}+ xy and x^{2} +2x + 3xy. It does not mean that a quadratic trinomial always turns into a quadratic equation when we equate it to zero. A quadratic equation is a quadratic trinomial formula with only one variable.
What is a Cubic Trinomial?
A cubic trinomial is a trinomial which has degree 3. For example, x^{3} +2x + 4 and 2y^{3}− 3y^{2} +1.
What is the Formula to Factor a Trinomial?
The factoring trinomials formulas of perfect square trinomials are:
 a^{2} + 2ab + b^{2} = (a + b)^{2}
 a^{2}  2ab + b^{2} = (a  b)^{2}
For applying either of these formulas, the trinomial should be one of the forms a^{2} + 2ab + b^{2} (or) a^{2}  2ab + b^{2}.
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