Trinomial
In short: A trinomial is a polynomial with exactly three terms, such as 2x² + 7x + 3, and the quadratic kind often factors into two binomials.
Algebra 1AlgebraA-APR.A.1
What is Trinomial?
With the prefix tri- meaning three, a trinomial is a polynomial having exactly three unlike terms, for example \(2x^2 + 7x + 3\) or \(5a^3 – 2a + 7\).
The most common one in Algebra 1 is the quadratic trinomial \(ax^2 + bx + c\). Many of them factor into a product of two binomials: \(2x^2 + 7x + 3 = (2x + 1)(x + 3)\), which you can verify by multiplying the factors back together.
How to spot it
- An expression has three terms in a row, separated by two plus or minus signs.
- The expression has the form \(ax^2 + bx + c\) and the instructions say to factor it.
- You need to add or subtract two polynomials, each holding three terms.
- A multiplication of two binomials produced three terms after like terms combined.
The method
- Count the terms; there must be three unlike terms, so merge like terms first.
- To add or subtract, line up like terms and combine their coefficients, flipping all signs when subtracting.
- To factor \(ax^2 + bx + c\), look for two numbers whose product is \(ac\) and whose sum is b.
- Multiply the factors back out to confirm you return to the original.
Trinomial: solved example
Problem
Add the trinomials \((x^2 + 4x – 5) + (3x^2 – x + 8)\) and say whether the result keeps three terms.
Solution
Group like terms: \(x^2 + 3x^2 = 4x^2\), \(4x – x = 3x\), and \(-5 + 8 = 3\).
The sum is \(4x^2 + 3x + 3\), which has three terms, so it is again a trinomial.
Try it
Question 1
How many terms does \(5a^3 – 2a + 7\) have, and what name fits?
Show the answer
Three terms, so it is a trinomial.
Question 2
Subtract: \((x^2 + 2x + 1) – (x^2 – 3x + 4)\).
Show the answer
\(x^2 + 2x + 1 – x^2 + 3x – 4 = 5x – 3\). Only two terms remain.
Trinomial vs. Binomial
The count of terms is the whole difference: two for a binomial, three for this one. Note that sums and differences can change the count, as the second question above shows: the \(x^2\) terms cancel and the trinomials leave a binomial behind.
Keep going
Lessons
Worksheets and quizzes
- Adding, Subtracting and Multiplying Polynomials – Worksheet
- Adding, Subtracting and Multiplying Polynomials – Quiz
- Factoring Polynomials – Worksheet
- Factoring Polynomials – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Algebra