完全平方三项式

Factoring Perfect Square Trinomials

Before we dive into this topic, it's important that break it down, and we are totally comfortable with two terms: perfect square and trinomial.

Let's start with trinomial.

Recall that a trinomial (a type of polynomial) is an expression composed of three terms that are joined together by addition or subtraction.

Also, recall that a perfect square is a term that results from multiplying another term with itself. For instance, $9$ is a perfect square because it is the product of $3$ and $3$.

So, after all of that, we can infer that a perfect square trinomial is an expression that is 1) composed of three terms that are joined together by addition or subtraction and 2) is the result of multiplying another expression with itself.

Identifying perfect square trinomials is pretty straightforward, as they'll always be in one of two forms: $$a^2 + 2ab + b^2\,\,\,\,\text{or}\,\,\,\,a^2-2ab+b^2.$$

These quadratics will factor into binomials with the form: $$(a + b)^2\\ \text{OR}\\ (a-b)^2$$

You can check your answers by FOILing.

Perfect square trinomials are essentially squared binomials that have been expanded (you can review how to square binomials here).

In other words, whenever you multiply out a squared binomial and simplify the result, you'll end up with a perfect square trinomial.

Some examples are the following:

• $(2x+3)^2 = 4x^2 + 12x + 9$
• $(x-5)^2 = x^2 - 10x + 25$

So, if you're asked to factor a perfect square trinomial, all you need to do is collapse that trinomial back into its squared binomial form.

For instance, as is shown in the first example above, the factored (or collapsed) form of the perfect square trinomial $4x^2+12x+9$ is $\left(2x+3\right)^2$.

Notice that factoring a perfect square trinomial of the form $a^2 + 2ab + b^2$ gives us an expression of the form $(a + b)^2$, and that factoring a perfect square trinomial of the form $a^2 - 2ab + b^2$ gives us an expression of the form $(a - b)^2$.

Now, let's reinforce the concepts we've learned about by working through a few examples.

So, as we can see, perfect square trinomials are essentially expanded squared binomials. This concept will show up a lot later and being able to recognize these trinomials will make complicated factoring much easier. In the next few lessons, you will learn a few more tricks and patterns for factoring polynomials.

(Video) Factoring Perfect Square Trinomials

by PatrickJMT

The video by PatrickJMT gives us some examples of how to factor perfect square trinomials.

$\large{\text{Summary}}$

The formulas for perfect square trinomials are: $$\begin{align} a^2+2ab+b^2&=(a+b)(a+b)\\ a^2-2ab+b^2&=(a-b)(a-b) \end{align}$$

Example: $\\$ $$9x^2+24xy+16y^2$$

We can rewrite this equation to be: $$9x^2+24xy+16y^2=(3x)^2+2(3x)(4y)+4y)^2$$

So we see that the formula can be used. This means that $$(3x)^2+2(3x)(4y)+4y)^2=(3x+4y)(3x+4y)$$

Model Factoring

Perfect square trinomials are in the form $x^2+2xy+y^2$ or $x^2-2xy+y^2$. These can be factored as $(x+y)^2$ and $(x-y)^2$ respectively. In other words, perfect square trinomials are the result of multiplying a binomial by itself (or squaring a binomial).

You can also think of it as finding the common factor of a trinomial, except the factors are two identical binomials.

Example 1: Let's look at a model factoring the expression $x^2+2x+1$.

Factor $x^2 + 2x+ 1$. We can rewrite $x^2+2x+1$ as $x^2+2x+1^2$. We can guess that the factorization is $(x+1)^2$, but we have to check the middle term $2x$. Expanding $(x+1)^2$, we get that $(x+1)^2 = x^2+2\cdot1\cdot x+1^2$, so we were correct. $x^2+2x+1 = (x+1)^2$.

Example 2: Let's look at a model factoring the expression $x^2+4x+4$.

Factor $x^2 + 4x+ 4$. We can rewrite $x^2+4x+4$ as $x^2+4x+2^2$. We can guess that the factorization is $(x+2)^2$, but we have to check the middle term $4x$. Expanding $(x+2)^2$, we get that $(x+2)^2 = x^2+4\cdot1\cdot x+2^2$, so we were correct. $x^2+4x+4 = (x+2)^2$.