Unit 1

Functions

Using Factors to Determine x-intercepts

Now, consider the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»7«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» discussed in the previous example. Because it is a polynomial function of degree «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math», its graph has the potential for three «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts. The fact the function has three unique binomial factors confirms it has three zeros, and the graph of the function has three «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.

The zeros of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» occur at:


In the event a polynomial does not have a constant term from which to identify possible integral zeros, consider first factoring the variable from all terms. For example, the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»8«/mn»«mi»x«/mi»«/mstyle»«/math» does not have a constant term. However, by factoring as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»x«/mi»«mfenced»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»8«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math», the constant term of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» can be used to identify possible integral factors of the resulting trinomial.