[br]In previous lessons, you learned how to factor special binomials—the [color=#0000ff]Difference of Two Squares (DOTS)[/color], the [color=#0000ff]Sum of Two Cubes (SOTC)[/color], and the [color=#0000ff]Difference of Two Cubes (DOTC)[/color].[br][br][i][color=#0000ff]Betting on Squares:[/color][/i] https://www.geogebra.org/m/d4jth98g[br][i][color=#0000ff]Shaking the Cubes:[/color] [/i]https://www.geogebra.org/m/xkbagzf4[br][br]In this lesson, you'll learn how to factor the only special trinomial—the [color=#0000ff]Perfect Square Trinomial (PST).[br][br][/color]A trinomial is a [color=#0000ff]Perfect Square Trinomial[/color] or transformable to one if:[br][br]1. There are [color=#0000ff]two squared terms[/color] preceded by [color=#0000ff]plus (+) signs[/color].[br]2. The remaining term is [color=#0000ff]twice[/color] [color=#0000ff]the product[/color] of the [color=#0000ff]square roots of the squared terms[/color]. Most pretenders (looking like PST, but are not) fail the second part of the test, particularly the [color=#0000ff]"twice"[/color] condition. The test is discussed in a previous lesson.[br][br][i][color=#0000ff]Going with the Flow:[/color][/i] https://www.geogebra.org/m/gtjh795t[br][br]The [color=#0000ff]Perfect Square Trinomial[/color] is factored as either a sum or a difference, depending on the sign of the middle term:[br][br][color=#0000ff]x² + 2xy + y² = (x + y)(x + y) = (x + y)² [/color]or[br][color=#0000ff]x² - 2xy + y² = (x - y)(x - y) = (x - y)²[/color].[br][br]Take note of the similarities and differences of the factors in each case.

[br]After the GCF has been factored out, you'll be asked to factor the quadratic as the product of two linear factors. [br][br]Let's review the conditions to determine if the resulting quadratic is a [color=#0000ff]Perfect Square Trinomial[/color]. Make sure you understand how to apply the test to any trinomial. [br][br]1. There must be [color=#0000ff]three terms[/color].[br]2. The [color=#0000ff]first[/color] and [color=#0000ff]third[/color] terms must be[color=#0000ff] perfect squares[/color] (1, 4, 9, 16, 25, ...) and the variables must have [color=#0000ff]even[/color] powers (2, 4, 6, 8, 10, ...).[br]3. The middle term must be [color=#0000ff]twice the product of the square roots of the first and third terms[/color].[br][br]If all conditions are met, you've got a Perfect Square Trinomial, and the factors can be written as the sum or difference of the square roots of the first and third terms, written twice or raised to the second power.[br][br]Example: 7a⁴ + 28a³ + 28a² = 7a²(a² + 4a + 4)[br][br]Let's determine if a² + 4a + 4 is a Perfect Square Trinomial.[br][br]1. There are three terms: a², 4a, and 4.[br]2. The first and third terms are perfect squares: a² and 4.[br]3. The middle term is twice the product of the square roots of the first and third terms: 2(a)(2) = 4a. The quadratic is a perfect square trinomial whose factors are (a + 2)(a + 2) or (a + 2)².[br][br]The complete factorization of the original problem is 7a²(a + 2)(a + 2) or 7a²(a + 2)².

[br]Below is a set of problems that require you to factor [color=#0000ff]Perfect Square Trinomials[/color].[br][br]Check the Answer Box below for the correct answers.

In future lessons, you'll learn how to factor general trinomials. Did you ENJOY our lesson today?