Key takeaway: A trinomial is a polynomial with exactly three unlike terms, such as 2x² − 5x + 7, joined by addition or subtraction.
Grade 9 – Algebra 1A-APR.A.1Expressions and Equations
Definition of Trinomial
A trinomial has three terms that cannot be combined with each other. The expressions \(x^{2} + 5x + 6\), \(4a – 3b + 9\) and \(m^{2} – 2mn + n^{2}\) all qualify. The prefix “tri” means three.
The most common one in Algebra 1 is the quadratic form \(ax^{2} + bx + c\), which describes parabolas and is the main target of factoring. For instance, the height of a ball thrown from a rooftop is often modeled by three terms: a squared time term, a time term and a starting height. Some are special: the expression \(x^{2} + 10x + 25\) comes from squaring a binomial, and it is a perfect square. When you multiply two binomials, the product often has three terms after like terms are combined.
Formula and notation
The usual single-variable pattern is
\[ax^{2} + bx + c, \qquad a \neq 0\]
Here \(a\) is the leading coefficient, \(b\) is the coefficient of the middle term and \(c\) is the constant. Write the terms from the highest power to the lowest, keeping each term’s sign.
Key facts
- Three terms; if two of them merge, only two remain.
- The most common form in Algebra 1 is ax² + bx + c.
- Evaluate by substituting a value and following the order of operations.
- Many of them factor into two binomials, which is the reverse of multiplying them.
Trinomial example
Evaluate the trinomial \(2x^{2} – 5x + 7\) when \(x = -3\).
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Substitute with parentheses: \(2(-3)^{2} – 5(-3) + 7\).
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Square first: \((-3)^{2} = 9\), so \(2 \times 9 = 18\).
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Multiply and add: \(18 + 15 + 7 = 40\).
The value is 40.
Common mistake
Watch out
Omar reads \((-3)^{2}\) as \(-9\) and gets \(2(-9) – 5(-3) + 7 = -18 + 15 + 7 = 4\).
Why it is wrong
The parentheses show that the whole number \(-3\) is squared, and a negative times a negative is positive: \((-3)^{2} = 9\). Only \(-3^{2}\), without parentheses, equals \(-9\). The correct total is \(18 + 15 + 7 = 40\).
Related terms and lessons
- Algebra 1 · Adding, Subtracting and Multiplying PolynomialsLessonWorksheetQuiz
- Algebra 1 · Factoring PolynomialsLessonWorksheetQuiz
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