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. x2−12x+36x^2 - 12x + 36
    2. 3x2−30x+753x^2 - 30x + 75
    3. −50x2+40xy−8y2-50x^2 + 40xy - 8y^2
  2. Find the square of a binomial:
    1. (−x−4y)2(-x - 4y)^2
    2. (−3x2+3y2)2(-3x^2 + 3y^2)^2
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Topic Notes
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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: x2+14x+49x^{2}+14x+49. Is this a perfect square trinomial? Looking at the first term, we've got x2x^{2}, which is a square. The last term is 4949. It is also a square since when you multiply 77 by 77, you'll get 4949. Therefore 4949 can also be written as 727^{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 14x14x when we have x2x^{2}and 727^{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 22. In other words, the perfect square trinomial formula is:

a2±ab+b2a^{2} \pm ab + b^{2}. We're now trying to see if we can get the middle term of 2ab2ab.

Since we've got our aa term as xx, and our bb term as 77, our 2ab2ab becomes 2∙7∙x2 \bullet 7 \bullet x. That gives us a total of 14x14x, which is the middle term in x2+14x+49x^{2}+14x+49! Therefore, we can rewrite the question as (x+7)2(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

x2−2x+36x^{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(x - 6)^{2}

Question 2:

Factor the perfect square

3x2−30x+753x^{2} - 30x +75

Solution:

Take out the common factor 33

3(x2−10x+25)3(x^{2} - 10x + 25)

Factor the x2−10x+25x^{2} - 10x + 25 and get the final answer:

3(x−5)23(x - 5)^{2}

Question 3:

Find the square of a binomial:

(−3x2+3y2)2(-3x^{2} + 3y^{2})^{2}

Solution:

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

ax2−bxy+cy2ax^{2} - bxy +cy^{2}

So the first term:

Square of −3x2=9x4-3x^{2} = 9x^{4}

The third term:

3y2=9y43y^{2} = 9y^{4}

The middle term is the multiplication of original 1st1^{st} and 2nd2^{nd} term, and then times 22

−3x2∙3y2=−9x2y2-3x^{2} \bullet 3y^{2} = -9x^{2}y^{2}

Then times 22:

−18x2y2-18x^{2}y^{2}

So the final answer:

(9x4−18x2y2+9y4)(9x^{4} - 18x^{2}y^{2} + 9y^{4})

To double check your answers, this online calculator will help you factor a polynomial expression. Use it as a reference, but make sure you learn how to properly go through the steps to answering a perfect square trinomial question.

Wasn't quite sure on the concepts covered in this chapter? Perhaps you may want to go back and review how to find common factors of polynomials or how to factor by grouping. Also read up on solving polynomials with unknown coefficients, and the intro to factoring polynomials.

Ready to move on? Up next, learn how to complete the square and change a quadratic function from standard form to vertex form.