L1.2 A2 Polynomial Functions - Part 2
Completion requirements
Unit 1
Functions
Polynomial functions are made of one or more terms. Terms are separated by addition or subtraction signs. Terms are composed of numerical coefficients and variables (of varying whole number degrees). A constant term is a term without a variable. The constant term remains unaffected by any variable values substituted into the function.
Consider the polynomial function «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»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»7«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
It has four terms. The first term, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mstyle»«/math», has a numerical coefficient of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» and a variable, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», of degree three.
The degree of the term is the sum of the exponents of the variable(s) in the term. Usually, polynomials are written in descending order by degree. The term of the highest degree is written first, followed by the term with the next highest degree, and so on. The constant term is written last.
The degree of the function is equal to the degree of the highest degree term.
Note:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»1«/mn»«/msup»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», so the degree is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» for the term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«mi»x«/mi»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»0«/mn»«/msup»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math», so the degree is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» for the constant term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»7«/mn»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»7«/mn»«msup»«mi»x«/mi»«mn»0«/mn»«/msup»«mo»=«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mfenced»«mn»1«/mn»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
Consider the polynomial function «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»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»7«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
It has four terms. The first term, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mstyle»«/math», has a numerical coefficient of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» and a variable, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», of degree three.
The degree of the term is the sum of the exponents of the variable(s) in the term. Usually, polynomials are written in descending order by degree. The term of the highest degree is written first, followed by the term with the next highest degree, and so on. The constant term is written last.
The degree of the function is equal to the degree of the highest degree term.
Note:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»1«/mn»«/msup»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», so the degree is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» for the term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«mi»x«/mi»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»0«/mn»«/msup»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math», so the degree is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» for the constant term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»7«/mn»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»7«/mn»«msup»«mi»x«/mi»«mn»0«/mn»«/msup»«mo»=«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mfenced»«mn»1«/mn»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«/mstyle»«/math»