A trinomial is a polynomial with exactly three terms, such as x2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax2 + bx + c, which students learn to factor into two binomial factors.
A trinomial has three unlike terms. The area model shows why x2 + 5x + 6 factors as (x + 2)(x + 3): the four pieces add to x2 + 5x + 6.
Formal definition and notation
A trinomial is a polynomial that, after like terms are combined, has exactly three nonzero terms, each a coefficient times a product of variables raised to whole-number powers. A quadratic trinomial in one variable has the form ax2 + bx + c with a, b, c all nonzero. Its degree is the highest degree among its terms.
Polynomial is the family name for any number of terms. Every trinomial is a polynomial, but x3 + x2 + x + 1, with four terms, is a polynomial that is not a trinomial.
A trinomial is an expression with no equals sign; x2 + 5x + 6 = 0 is an equation you solve. Factoring the trinomial is one way to solve it.
Quick check: 3 questions
Question 1 of 3
Which expression is a trinomial?
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Answer: C.x2 − 6x + 9 has three unlike terms. 5x2 − 3 has two, 4x + 2x − 7 simplifies to 6x − 7, and the last one has four terms.
Question 2 of 3
Factor x2 − 9x + 20.
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Answer: A. You need a product of +20 and a sum of −9: −4 and −5. (x + 4)(x + 5) gives +9x, (x − 2)(x − 10) gives −12x, and (x + 4)(x − 5) gives a constant of −20.
Question 3 of 3
Which trinomial is a perfect square?
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Answer: D.x2 + 14x + 49 = (x + 7)2, since 2 × 7 = 14 and 72 = 49. x2 + 5x + 25 would need 10x, x2 + 6x + 8 is (x + 2)(x + 4), and x2 + 9 is a binomial.
Trinomial: frequently asked questions
What is an example of a trinomial?
x2 + 5x + 6 is the classic example: a squared term, an x term and a constant. Others are 3y2 − y + 10, a + b + c and x4 − 2x2 + 1. Each has exactly three terms that are not like terms, so none of them can be combined into fewer terms.
How do you factor a trinomial?
For x2 + bx + c, find two numbers that multiply to c and add to b, then write (x + p)(x + q). For x2 + 8x + 15, 3 and 5 work, so it factors as (x + 3)(x + 5). When the leading coefficient is not 1, multiply a × c, split the middle term with the pair you find, and factor by grouping.
Can every trinomial be factored?
No. Many trinomials have no factors with whole-number coefficients. x2 + x + 1 is one: no two integers multiply to 1 and add to 1. Such trinomials are called prime over the integers. The quadratic formula can still find the solutions of x2 + x + 1 = 0, but they are not real numbers.
PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Standard Form of a QuadraticThe standard form of a quadratic is ax2 + bx + c, with the terms in descending powers of x and a ≠ 0. As an equation it is ax2 + bx + c = 0; as a function, y = ax2 + bx + c. It shows a, b and c directly, which the quadratic formula needs, and c is the y-intercept.Algebra · Algebra 1, Algebra 2Perfect SquareA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and √49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Number System · Grade 6, Grade 7, Grade 8Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Leading CoefficientThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x2 − 5x4 + 2x, the highest power is x4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph’s ends point.Algebra · Algebra 1, Algebra 2Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax2 + bx + c = 0 with a ≠ 0: x = −b ± √b2 − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Completing the SquareCompleting the square is a method that rewrites x2 + bx as a perfect square by adding (b2)2: x2 + bx + (b2)2 = (x + b2)2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra · Algebra 1, Algebra 2, PrecalculusDegree of a PolynomialThe degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x3 − 2x5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x2y4 has degree 6. A nonzero constant such as 9 has degree 0.Algebra · Algebra 1, Algebra 2Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8