The relationship you explored in the Investigation can be seen by squaring the binomial «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/math».

Notice that the terms in the trinomial are related to the terms in the binomial, as shown below.



Alternatively, we can look at a perfect square trinomial in the opposite way.



According to this pattern, any trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mi»y«/mi»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/math» can be factored as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/math».
 

Example 1

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«mi»x«/mi»«mo»+«/mo»«mn»36«/mn»«/math».

The first term of the trinomial is the perfect square «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«/math», and the third term of the trinomial is the perfect square «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»(«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»6«/mn»«msup mathcolor=¨#B94A48¨»«mo mathcolor=¨#B94A48¨»)«/mo»«mn»2«/mn»«/msup»«/math». These two observations are enough to further investigate the possibility that this is a perfect-square trinomial.

The middle term of the trinomial is twice the product of the square root of the first and third terms. Therefore, the trinomial is a perfect-square trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mn»2«/mn»«/msup»«/math».

 When the second term of the trinomial is negative, it indicates that the binomial factors must contain a negative as well.

This trinomial could also have been factored as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mo»-«/mo»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/math». If you are uncertain, expand both «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mo»-«/mo»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mi»x«/mi»«mo»-«/mo»«mn»6«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/math» and note that each expansion produces the trinomial «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«mi»x«/mi»«mo»+«/mo»«mn»36«/mn»«/math».


Example 2

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»24«/mn»«mi»x«/mi»«mo»+«/mo»«mn»16«/mn»«/math».

The first term of the trinomial is the perfect square «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»3«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«msup mathcolor=¨#B94A48¨»«mo mathcolor=¨#B94A48¨»)«/mo»«mn»2«/mn»«/msup»«/math», and the third term of the trinomial is the perfect square «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#B94A48¨»«mn mathcolor=¨#B94A48¨»4«/mn»«mn»2«/mn»«/msup»«/math». These two observations are enough to further investigate the possibility that this is a perfect-square trinomial. The middle term of the trinomial is twice the product of the square root of the first and third terms. This trinomial is a perfect-square trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mn»2«/mn»«/msup»«/math».


Recognizing that a trinomial is a perfect-square trinomial can lead to a quick factorization. However, it is still possible to use algebra tiles or decomposition to factor these perfect-square trinomials.