Unit 7A

Integrals Part 1

Lesson 4: Areas Part 2

Skill Builder

Polynomial Functions

Considering the end behaviour of a graph is a good way to help picture a function’s graph.

Determining what the left endpoint of a graph is doing, or where it is extending, and then doing the same for the right endpoint of a graph will help you visualize the graph of a function.

Two things to consider when thinking about a graph’s end behaviour are the degree of the polynomial and the sign of the leading coefficient.

As seen in the chart below, the degree of a function determines the general shape of the function’s graph.

The sign of the leading coefficient will help you determine the end behaviour of the function.

Then, by determining specific points, such as the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts and the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercepts, more details about the shape of the graph are revealed.

The Factor Theorem

The factor theorem is useful for determining whether a binomial is a factor of a polynomial.

The factor theorem states that «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/math» is a factor of a polynomial, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», if and only if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mfenced»«mi»a«/mi»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/math».

In other words, if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is substituted for the variable in a polynomial function, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», and the result is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«/math», then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/math» is a factor of the polynomial «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math».

Simply put, the factor theorem is a special case of the remainder theorem, for which the remainder is always «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«/math».

Possible Integral Zeros
When factoring a polynomial, instead of randomly choosing a value for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» that might work, narrow the selection process by considering only a specific list of possible integers that, when substituted into the polynomial function, could result in a function value of zero. To do this, use the integral zero theorem, which states that if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/math» is a factor of a polynomial function with integral coefficients, then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is a factor of the constant term.


Using Factors to Determine «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»x«/mi»«/math»-intercepts
Now, consider the function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«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»«/math» discussed in the previous example. Because it is a polynomial function of degree «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«/math», its graph has the potential for three «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/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 xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»-intercepts.

The zeros of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/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 xmlns=¨http://www.w3.org/1998/Math/MathML¨»«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»«/math» does not have a constant term. However, by factoring as «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«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»«/math», the constant term of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«/math» can be used to identify possible integral factors of the resulting trinomial.