A trinomial is a type of polynomial that consists of three terms. These terms are usually written as ax² + bx + c, where a, b, and c are constants, and x is the variable.
Other examples:
- x2 + 5x + 6
- 2a2 − 3a + 1
- 15x2 - 42x + 3z
- xyz3 - z2 + y3
Polynomial Types: Monomial, Binomial, and Trinomial
| Number of terms | Polynomial | Example |
|---|---|---|
| 1 | Monomial | 3x2y |
| 2 | Binomial | 4x - y |
| 3 | Trinomial | 3x2 + 2x - 1 |
Perfect Square Trinomial
A perfect square trinomial is a mathematical expression formed by squaring a binomial.
- It follows the pattern ax2 + bx + c, where (a), (b), and (c) are real numbers, and (a) is not equal to zero.
- It also meets the condition b2 = 4ac.
The perfect square trinomial formula is
- (ax)2 + 2abx + b2 = (ax + b)2
- (ax)2 - 2abx + b2 = (ax - b)2
For instance, consider the binomial (x + 2)2. When expanded, it results in x2 + 4x + 4, which is a perfect square trinomial. To decompose a perfect square trinomial, one can express it as the product of two identical binomials.
For example, x2 + 4x + 4 can be factored as (x + 2)(x + 2). When you multiply (x + 2) with itself, it will obtain x2 + 4x + 4, confirming that the expression is a perfect square trinomial.
Identifying a Perfect Square Trinomial
To recognize a perfect square trinomial, follow these steps:
Check Form: Look at the expression, and if it's in the form (ax2 + bx + c), it could be a perfect square trinomial.
Verify Condition: Confirm if the condition b2 = 4ac is met. Here, (b) is the coefficient of the linear term, and (a) and (c) are the coefficients of the squared and constant terms, respectively.
Compare with Formula: See if the expression matches the structure of (ax + b)2 or (ax - b)2. If it does, then it's a perfect square trinomial.
Quadratic Trinomial
A quadratic trinomial is a specific kind of mathematical expression containing both variables and constants. It appears in the form (ax2 + bx + c), where x (x) is the variable, and (a), (b), and (c) are real numbers that are not zero. Here, (a) is called the leading coefficient, (b) is the linear coefficient, and (c) is the additive constant.
There's a key aspect related to quadratic trinomials, called the discriminant (D), expressed as (D = b2 - 4ac). The discriminant helps categorize different cases of quadratic trinomials. By evaluating (D), you can understand more about the nature of the quadratic expression.
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Identities of Trinomials
Trinomial identities refer to formulas involving algebraic expressions with three terms.
- Factorization Identity: (a + b)(a + c)(b + c) = (a + b + c)(ab + ac + bc) − abc
- Square Sum Identity: a2 + b2 + c2 = (a + b + c)2 − 2(ab + ac + bc)
- Sum of Cubes Identity: a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2− ab − ac − bc)
Factoring Trinomials
Factoring trinomials means expressing a polynomial with three terms as a product of two binomials.
A trinomial is usually written as ax2+bx+c, where a, b, and c are constants.
Steps to Factor Trinomials
Factoring trinomials is a process of expressing a mathematical expression with three terms as a product of two binomials
Step 1: Recognize if the trinomial is in the quadratic form (ax2 + bx + c).
Step 2: Start by factoring out the greatest common factor among the coefficients of (a), (b), and (c).
Step 3: Look for patterns such as perfect squares or the difference of squares, which might simplify the factoring process.
Step 4: If the trinomial has four terms, group them and factor out common terms from each group.
Step 5: For trinomials in the form (ax2 + bx + c), use methods like trial and error, grouping, or the AC method to factor.
Step 6: Verify your factoring by multiplying the binomials to ensure they correctly expand back to the original trinomial
Quadratic Trinomial in One Variable
The quadratic trinomial formula in one variable is given by (ax2 + bx + c), with (a), (b), and (c) being constant terms, and none of them is zero. If the value of b2 - 4ac is greater than zero (b2 - 4ac > 0), then we can always express the quadratic trinomial as a product of two binomials: a(x + h)(x + k), where (h) and (k) are real numbers.
Example: Factorize 2x2 - 7x + 3
Here, a = 2, b = -7, and c = 3.
Calculate b2 - 4ac: (-7)2 - 4(2)(3) = 49 - 24 = 25
b2 - 4ac = 25 > 0
Using Quadratic Formula,
2x2 - 7x + 3 = 2(x + 1/2)(x - 3)
Quadratic trinomial 2x2 - 7x + 3 can be factorized as 2(x + 1/2)(x - 3)
Quadratic Trinomial in Two Variable
The general form of a quadratic trinomial in two variables is (ax2 + bxy + cy2 + dx + ey + f)
where (a), (b), (c), (d), (e), and (f) are constant terms, and at least one of (a), (b), or (c) is nonzero. A quadratic trinomial in two variable is factorized as,
Example: Factorize quadratic trinomial x2 - 4xy + 4y2
= x2 - 4xy + 4y2
= x2 + (2y)2 - 2(x)(2y)
= (x - 2y)2
Quadratic trinomial x2 - 4xy + 4y2 can be factorized as (x - 2y).(x - 2y)
Factoring Trinomial by Splitting Middle Term
Factoring a trinomial by splitting the middle term is a method used when factoring quadratic trinomials in the form (ax2 + bx + c), where (a), (b), and (c) are constants, and (a) is not equal to zero. The goal is to split the middle term (bx) into two terms whose coefficients multiply to give (a × c).
Steps to factorize a trinomial by splitting the middle term are:
Step 1: Identify the coefficients (a), (b), and (c) from the trinomial.
Step 2: Calculate the product of (a) and (c).
Step 3: Break the middle term (bx) into two terms whose coefficients multiply to give (a × c). Rewrite the trinomial accordingly.
Step 4: Group the terms into pairs and factor out the common factor from each pair.
Step 5: Factor out the common binomial factor from the grouped terms.
Example: Factorize (x2 - 5x + 6)
Coefficient of a=1, b=-5, c=6
Find (a × c) = 1 × 6 = 6
Split middle term: Rewrite (-5x) as (-2x - 3x), where (-2 × -3 = 6)
Factor by grouping: (x2 - 2x - 3x + 6)
Factor out common binomial factor: x(x - 2) - 3(x - 2)
(x2 - 5x + 6) factors as (x - 2)(x - 3)
Trinomial Identity
If a trinomial is an identity, it means it can be factored into the product of two binomials. An identity in this context is a mathematical expression that remains true for all values of the variable. In the case of factoring trinomial identities, the goal is to find the equivalent form of the trinomial as the product of two binomials.
Consider an example: x2 + 6x + 9
Identify Pattern
Identify whether the trinomial follows the pattern of a perfect square trinomial
In this example, x2 + 6x + 9 matches the pattern a2 + 2ab + b2
where a = x and b = 3
Apply Perfect Square Trinomial Identity
Use identity (a + b)2 = a2 + 2ab + b2
x2 + 6x + 9 = (x + 3)2
Some identity that are used to solve trinomial identity are,
Identity | Expanded Form |
|---|---|
(x + y)2 | x2 + 2xy + y2 |
(x - y)2 | x2 - 2xy + y2 |
(x2 - y2) | (x + y)(x - y) |
Factorizing with GCF
Factorizing with GCF (Greatest Common Factor) is a method used when a quadratic trinomial has more than one term, and the terms share a common factor. The idea is to factor out this common factor to simplify the expression.
Step 1: For quadratic trinomial (ax2 + bx + c), identify the coefficients (a), (b), and (c)
Step 2: Determine the Greatest Common Factor of (a), (b), and (c)
Step 3: Divide each term of the trinomial by the GCF and factor it out
Step 4: Write the factored form as the product of the GCF and the remaining expression
Example: Factorize quadratic trinomial (6x2 + 9x + 3)
Coefficients are: (a = 6), (b = 9), (c = 3)
GCF of 6, 9, and 3 is 3
Factored form is 3(2x2 + 3x + 1)
So, quadratic trinomial (6x2 + 9x + 3) can be factorized as 3(2x2 + 3x + 1) using the GCF method.
Trinomials with Leading Coefficient of 1
Example: Factorize quadratic trinomial (x2 + 5x + 6)
Coefficients are (a = 1), (b = 5), and (c = 6)
Trinomial is of form (ax2 + bx + c) where (a) is 1
To factorize it, we need to find two numbers whose sum is (b) (5 in this case) and whose product is (ac) (product of (a) and (c), which is (1 × 6 = 6)
i.e. (2 + 3 = 5) and (2 × 3 = 6)
x2 + 5x + 6 = x2 + 2x + 3x + 6
(x2 + 2x) + (3x + 6)
x2 + 2x = x(x + 2) and 3x + 6 = 3(x + 2)
x(x + 2) + 3(x + 2)= (x + 2)(x + 3)
Factored form of (x2 + 5x + 6) is (x + 2)(x + 3)
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Solved Examples
Example 1: Factor the trinomial: (y2 - 6y + 9)
Coefficients of a, b, and c are
- a=1
- b= -6
- c=9
(1 × 9 = 9) and 1 + (-6) = -5)
= y2 - 3y - 3y + 9
= y(y - 3) - 3(y - 3)
= (y - 3)(y - 3) or (y - 3)2
Example 2: Factor the trinomial l: 4m2 - 12m + 9
Coefficients of a, b, and c are
- a = 4
- b = -12
- c = 9
(4 × 9 = 36) and 4 + (-12) = -8)
= (4m2 - 6m - 6m + 9)
= 2m(2m - 3) - 3(2m - 3)
= (2m - 3)(2m - 3) or (2m - 3)2
Practice Questions
Q1. Factorize Trinomial: x2 + 7x + 12
Q2. Factorize Trinomial: 2x2 - 5x - 3
Q3. Factorize Trinomial: 3y2 + 10y + 7
Q4. Factorize Trinomial: 4a2 - 4a - 3
Q5. Factorize Trinomial: x2 - 6x + 9