B. Factoring Perfect-Square Trinomials

B. Factoring Perfect-Square Trinomials

When a binomial is squared, the trinomial produced is often called a perfect-square trinomial because of some special characteristics of the coefficients of both the x²-term and the x-term, and the constant term. Recognizing these characteristics will allow you to factor perfect-square trinomials very quickly. The following Investigation explores these characteristics.

Perfect-Square Trinomial a trinomial that is formed by squaring a binomial

Investigation

1. Multiply «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/math». Remember, «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»=«/mo»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mo»)«/mo»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mo»)«/mo»«/math».
   
   

2. Some other binomials have been squared to produce trinomials 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»«mi»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math», for your reference.
   
   
   
   For each of the examples provided,
   
   

1. Suggest a relationship between the binomial and the b-value of the trinomial.
   
   

2. Suggest a relationship between the binomial and the c-value of the trinomial.
   
   

3. For your reference, more binomials have been squared to produce trinomials of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math».
   
   
   
   For each of the examples provided,
   
   

1. Suggest a relationship between the binomial and the a-value of the trinomial.
   
   

2. Suggest a relationship between the binomial and the b-value of the trinomial.
   
   

3. Suggest a relationship between the binomial and the c-value of the trinomial.
   
   

4. Use these relationships to factor each of the following expressions.
   
   

1. 
   
   

2.