# Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2

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##### Intros
###### Lessons
1. Intro to Factoring Perfect Square Polynomials
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##### Examples
###### Lessons
1. Factor the perfect squares
1. $x^2 - 12x + 36$
2. $3x^2 - 30x + 75$
3. $-50x^2 + 40xy - 8y^2$
2. Find the square of a binomial:
1. $(-x - 4y)^2$
2. $(-3x^2 + 3y^2)^2$
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###### Topic Notes
Some polynomials have common patterns, which can be factorized faster if you can recognize them. Perfect square trinomial is one of these cases.

## What is a perfect square trinomial

Let's first remember what a trinomial is. A polynomial has several terms. A trinomial (as the prefix "tri-" suggests) is a polynomial with three terms. When we're dealing with perfect squares, it means we're dealing with squaring binomials. Continue on to learn how we go about factoring a trinomial.

## How to factor perfect square trinomials

One good way to recognize if a trinomial is perfect square is to look at its first and third term. If they are both squares, there's a good chance that you may be working with a perfect square trinomial.

Let's say we're working with the following: $x^{2}+14x+49$. Is this a perfect square trinomial? Looking at the first term, we've got $x^{2}$, which is a square. The last term is $49$. It is also a square since when you multiply $7$ by $7$, you'll get $49$. Therefore $49$ can also be written as $7^{2}$. The next step to identifying if we've got a perfect square is to see if we are able to get the middle term of $14x$ when we have $x^{2}$and $7^{2}$ to work with.

In the case of a perfect square, the middle term is the first term multiplied by the last term, and then multiplied by $2$. In other words, the perfect square trinomial formula is:

$a^{2} \pm ab + b^{2}$. We're now trying to see if we can get the middle term of $2ab$.

Since we've got our $a$ term as $x$, and our $b$ term as $7$, our $2ab$ becomes $2 \bullet 7 \bullet x$. That gives us a total of $14x$, which is the middle term in $x^{2}+14x+49$! Therefore, we can rewrite the question as $(x + 7)^{2}$through factoring perfect square trinomials. You've solved a perfect square trinomial! You're now ready to apply trinomial factoring to some practice problems.

## Example problems

Question 1:

Factor the perfect square

$x^{2} - 2x + 36$

Solution:

We know that this is a perfect square, and all we're asked is to factor it. Therefore, just take a look at the first and last term and find what they are squares of. It'll give us:

$(x - 6)^{2}$

Question 2:

Factor the perfect square

$3x^{2} - 30x +75$

Solution:

Take out the common factor $3$

$3(x^{2} - 10x + 25)$

Factor the $x^{2} - 10x + 25$ and get the final answer:

$3(x - 5)^{2}$

Question 3:

Find the square of a binomial:

$(-3x^{2} + 3y^{2})^{2}$

Solution:

You can square it and it will become what we have here:

$ax^{2} - bxy +cy^{2}$

So the first term:

Square of $-3x^{2} = 9x^{4}$

The third term:

$3y^{2} = 9y^{4}$

The middle term is the multiplication of original $1^{st}$ and $2^{nd}$ term, and then times $2$

$-3x^{2} \bullet 3y^{2} = -9x^{2}y^{2}$

Then times $2$:

$-18x^{2}y^{2}$

$(9x^{4} - 18x^{2}y^{2} + 9y^{4})$