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Trinomial

Trinomial

A polynomial with three terms which are not like terms. The following are all trinomials: x2 + 2x - 3, 3x5 - 8x4 + x3, and a2b + 13x + c.

 

 

 

See also

Monomial, binomial

Worked Example

Problem: Determine whether each expression is a trinomial: (a) 4x2+3x−104x^2 + 3x - 10, (b) x2+5x+2x+1x^2 + 5x + 2x + 1, (c) 7a3−2ab+97a^3 - 2ab + 9.
Step 1: Check expression (a): 4x2+3x−104x^2 + 3x - 10. Count the terms: 4x24x^2, 3x3x, and −10-10. That is three terms, and none are like terms.
4x2+3x−10⇒3 unlike terms — trinomial4x^2 + 3x - 10 \quad \Rightarrow \quad \text{3 unlike terms — trinomial}
Step 2: Check expression (b): x2+5x+2x+1x^2 + 5x + 2x + 1. It appears to have four terms, but 5x5x and 2x2x are like terms. Combine them to get x2+7x+1x^2 + 7x + 1.
x2+5x+2x+1=x2+7x+1⇒3 unlike terms — trinomialx^2 + 5x + 2x + 1 = x^2 + 7x + 1 \quad \Rightarrow \quad \text{3 unlike terms — trinomial}
Step 3: Check expression (c): 7a3−2ab+97a^3 - 2ab + 9. Count the terms: 7a37a^3, −2ab-2ab, and 99. That is three terms, and none are like terms.
7a3−2ab+9⇒3 unlike terms — trinomial7a^3 - 2ab + 9 \quad \Rightarrow \quad \text{3 unlike terms — trinomial}
Answer: All three expressions are trinomials. Expression (b) requires combining like terms first, but it simplifies to three terms.

Another Example

Problem: Factor the trinomial x2+7x+12x^2 + 7x + 12.
Step 1: Find two numbers that multiply to 12 (the constant term) and add to 7 (the coefficient of xx).
p⋅q=12andp+q=7p \cdot q = 12 \quad \text{and} \quad p + q = 7
Step 2: Test factor pairs of 12: 1×121 \times 12, 2×62 \times 6, 3×43 \times 4. The pair 33 and 44 gives 3+4=73 + 4 = 7.
p=3,q=4p = 3, \quad q = 4
Step 3: Write the trinomial as a product of two binomials using these values.
x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x + 3)(x + 4)
Answer: The factored form is (x+3)(x+4)(x + 3)(x + 4).

Frequently Asked Questions

Is x2+9x^2 + 9 a trinomial?
No. The expression x2+9x^2 + 9 has only two terms (x2x^2 and 99), so it is a binomial, not a trinomial. A trinomial must have exactly three unlike terms.
Can a trinomial have more than one variable?
Yes. A trinomial just needs exactly three unlike terms. For example, a2+2ab+b2a^2 + 2ab + b^2 is a trinomial with two variables. The number of variables does not matter — only the number of terms does.

Trinomial vs. Binomial

A trinomial has exactly three unlike terms (e.g., x2+5x+6x^2 + 5x + 6), while a binomial has exactly two unlike terms (e.g., x+3x + 3). Both are special types of polynomials, classified by their number of terms. Multiplying two binomials together often produces a trinomial, which is why factoring trinomials back into binomials is such a common algebra task.

Why It Matters

Trinomials appear constantly in algebra, especially quadratic trinomials of the form ax2+bx+cax^2 + bx + c. Learning to factor these is a core skill used to solve quadratic equations, simplify rational expressions, and analyze parabolas. Beyond quadratics, recognizing trinomial structure helps you apply special factoring patterns like perfect square trinomials.

Common Mistakes

Mistake: Counting terms before combining like terms and concluding the expression has more or fewer than three terms.
Correction: Always simplify by combining like terms first. For instance, x2+3x+2x+5x^2 + 3x + 2x + 5 looks like four terms but simplifies to the trinomial x2+5x+5x^2 + 5x + 5.
Mistake: Thinking a trinomial must have exactly three different variables.
Correction: The word 'tri' refers to three terms, not three variables. A single-variable expression like 2x3−x+42x^3 - x + 4 is still a trinomial because it has three unlike terms.

Related Terms