Mathematical investigation on the features of quartic polynomials and conjectures about the relationship between the number of turning points and inflection points. Includes graphs of polynomials, and investigation into all of the cases of quartic polynomials including all repeated roots, one cub...
STAGE 1 ASSESSMENT TYPE 2: INVESTIGATING THE FEATURES OF POLYNOMIALS
Introduction: A polynomial is a mathematical expression that has one or a greater number of terms, involving only the four basic
operations, and one or more variables. Polynomials have a wide range of applications in various fields, where they are used in the
modelling of simple to sophisticated phenomena such as projectile motion in physics, optimisation of technology, and modelling
anything from weather patterns, to chemical and biological processes (Weaser, 2007).
The degree of a polynomial is the highest power in the expression. Polynomials cannot have negative or fractional exponents, a
variable in the denominator, or a variable under a radical (MyMathEducation 2014).
This investigation will focus on analysing the features of real quartic polynomials of degree four as well as comparing them to
polynomials of other degrees. The aim of this investigation is to examine the relationship between the number of turning and
inflection points in quartic polynomials.
Features of polynomial graphs: There are several features of a polynomial function graph. The roots of a polynomial are all the
points where y=0. The turning points are points where the graph changes direction. The points of inflection are where the function
changes shape from concave up to concave down (or vice versa) (Cambridge Senior Methods 2017).
Method: To start, polynomials of degree four (quartic polynomials) with four different distinctive roots were graphed using
GeoGebra in a table (see table 1A). The roots, number of turning points and inflection points were recorded for each. To generate
these points on the graphs, GeoGebra commands were used including r=root (f) for the roots, e=extremum (f) for turning points,
and i=inflectionpoint (f) for the inflection points (shaunteaches 2012).
Table 1A: Degree four real polynomials with four distinct real linear factors
p( x) a( x )( x )( x )( x ), a 0
Function Roots used Number of Number of points of Graph
turning points inflection points
(x+1)(x+2)(x+3)(x+4) -4, -3, -2, -1 3 real 2 real
(x+0.5)(x+0.6)(x+0.7)(x+0.8) -0.8, -0.7, -0.6, - 3 real 2 real
0.5
(x-1)(x-2)(x-3)(x-4) 1, 2, 3, 4 3 real 2 real
(x-4)(x+6)(x+0.2)(x+2.4) 4, -6, -0.2, -2.4 3 real 2 real
-3(x-0.3)(x-0.5)(x+0.2)(x+1) -1, -0.2, 0.3, 0.5 3 real 2 real
2(x+4)(x+7)(x+6)(x+5) -7, -6, -5, -4 3 real 2 real
, STAGE 1 ASSESSMENT TYPE 2: INVESTIGATING THE FEATURES OF POLYNOMIALS
Interpretation: What all the investigated quartic polynomial functions have in common is they have the same number of turning
points (three) and inflection points (two). To find the relationship between the number of turning and inflection points for quartic
polynomials with different roots, polynomial expressions of degree five with five distinctive roots were tested as well to help
verify a trend (see Table 2A).
Table 2A: Degree five real polynomials with five distinct linear factors
𝑝(𝑥) = 𝑎(𝑥 − 𝛼)(𝑥 − 𝛽)(𝑥 − 𝛾)(𝑥 − 𝛿)(𝑥 − 𝜀), 𝑎 ≠ 0
Function Roots used Number of Number of points Graph
turning of inflection points
points
(x-1)(x-2)(x-3)(x-4)(x-5) 1, 2, 3, 4, 5 4 real 3 real
-0.2(x+2)(x-3)(x+4.5)(x- -4.5, -2, -1, 1.5 4 real 3 real
1.5)(x+1)
-(x+1)(x+5)(x+3)(x-2)(x-3) -5, -3, -1, 2, 3 4 real 3 real
Interpretation: These results also show that all the polynomials had the same number of turning and inflection points.
Conjecture: Relationship between the number of turning points and points of inflection for a polynomial of degree, n, with n
distinct real linear factors in the factorised form.
• Given the above data, the number of turning points seams to equal to the number of roots subtract one and the number of
inflection points equals to the number of roots subtract two, given no two roots are the same.
∴ 𝑡. 𝑝 = 𝑛 𝑟𝑜𝑜𝑡 − 1 ∶ 𝑖. 𝑝 = 𝑛 𝑟𝑜𝑜𝑡 − 2
To test this hypothesis, polynomials of higher degrees for n ( ) were analysed with the
information recorded below.
Example 1- This polynomial of degree six has six distinctive roots. Therefore, it will have five turning and four inflection points.
p(x) = 0.1 (x-1) (x+2) (x-2) (x+3) (x-4) (x+4)
The number of turning points in this function is equal to
6-1 (five turning points) and the number of inflection
points is equal to 6 – 2 (four inflection points).
Example 2- This polynomial degree seven has seven distinctive roots. Therefore, it will have six turning and five inflection points.
p(x)=0.1 (x-0.1) (x-0.2) (x-0.3) (x-0.4) (x-0.5) (x-0.6) (x-0.7)
The number of turning points in this function is equal to
7-1 (six turning points) and number of inflection points
is equal to 7 – 2 (five inflection points).
The above polynomials support the conjecture that n roots subtract one is the number of turning points and n roots subtract two is
the number of inflection points given no two roots are the same.
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