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Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares
The polynomial
Learning Objectives

Apply factoring techniques to factor a trinomial of the form ax2 + bx + c

Differentiate between perfect square trinomials and nonperfect square trinomials
Key Points
 Some trinomials, known as perfect square trinomials, can be factored into two equal binomials.
 We can factor
$a^2 b^2$ , the difference of two squares, by finding the terms that produce the perfect squares and substituting these quantities into the factorization form. When using real numbers, there is no factored form for the sum of two squares.  Perfect square trinomials factor as the square of a binomial. To recognize them, look for whether (1) the first and last terms are perfect squares, and (2) the middle term is divisible by 2, and when halved, equals the product of the terms that when squared produce the first and last terms.
Terms

trinomial
A polynomial expression consisting of three terms, or monomials, separated by addition and/or subtraction symbols.

binomial
A polynomial consisting of two terms, or monomials, separated by addition or subtraction symbols.
Full Text
Factoring Trinomials
The polynomial
Interactive Graph: Polynomial
Graph of a trinomial with the example equation
Ultimately, the trinomial should be factored in the form
Perfect Squares
Some trinomials, known as perfect square trinomials, can be factored into two equal binomials. For example:
and
Perfect square trinomials always factor as the square of a binomial.
To recognize a perfect square trinomial, look for the following features:
 The first and last terms are perfect squares.
 The middle term is divisible by 2.
For example,
Given that the coefficient of x^{2} is 1, we know that the factored form will be
Key Term Reference
 coefficient
 Appears in these related concepts: The LeadingTerm Test, Linear Combination of Atomic Orbitals (LCAO), and Balancing Chemical Equations
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 factor
 Appears in these related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 factorization
 Appears in this related concept: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1
 polynomial
 Appears in these related concepts: Polynomials: Introduction, Addition, and Subtraction, Partial Fractions, and Greatest Common Factor and Factoring by Grouping
 real number
 Appears in these related concepts: Graphing the Normal Distribution, Intermediate Value Theorem, and The ComplexNumber System
 real numbers
 Appears in these related concepts: Finding Domains of Functions, The Intermediate Value Theorem, and Linear Inequalities
 square
 Appears in these related concepts: Standard Form and Completing the Square, Special Factorizations and Binomials, and Radical Equations
 term
 Appears in these related concepts: Arithmetic Sequences, Basics of Graphing Polynomial Functions, and The 22nd Amendment
 variable
 Appears in these related concepts: Related Rates, Fundamentals of Statistics, and Math Review
Sources
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Cite This Source
Source: Boundless. “Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares.” Boundless Algebra. Boundless, 01 Jul. 2015. Retrieved 02 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/factoring12/factoringtrinomialsoftheformax2bxcperfectsquares725876/