This cluster includes the following access points.
Vetted resources educators can use to teach the concepts and skills in this topic.
| Name |
Description |
| Zeroes and factorization of a general polynomial: | In this task, students are asked to show or verify four theorems related to roots, zeroes, and factors of polynomial functions. The Fundamental theorem of Arithmetic is also mentioned. This task builds on "Zeroes and factorization of a quadratic function'' parts I and II. |
| Zeroes and factorization of a non polynomial function: | For a polynomial function f, if f(0)=0 then the polynomial f(x) is divisible by x. This fact is shown and then generalized in "Zeroes of a quadratic polynomial I, II" and "Zeroes of a general polynomial.'' Here, divisibility tells us that the quotient f(x)/x will still be a nice function -- indeed, another polynomial, save for the missing point at x=0. The goal of this task is to show via a concrete example that this nice property of polynomials is not shared by all functions. The non-polynomial function F given by F(x)=|x| is a familiar function for which property does not hold: even though F(0)=0, the quotient F(x)/x behaves badly near x=0. Indeed, its graph is broken into two parts which do not connect at x=0. |
| Zeroes and factorization of a quadratic polynomial II: | This task continues "Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in "Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. In the quadratic case, an alternative argument for why there can be at most two roots can be given using the quadratic formula and this is done in the second solution below.
This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions. Students who are familiar with the quadratic formula should be encouraged to think about the first solution which extends to polynomials of higher degree where formulas for the roots are either very complex or not possible to find. |
| Zeroes and factorization of a quadratic polynomial I: | For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x-r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. The teacher should pay close attention to the logic used in the solution to part (c) where the divisibility of ax2+bx+c by x-r is obtained not by performing long division but by using the result of long division of these polynomials; namely, that said division will result in an expression of the following form:
ax2+bx+c=(x-r)l(x)+k
where l is a linear polynomial and k is a number.
This task could be used either for assessment or for instructional purposes. If it is used for assessment, parts (a) and (b) are more suitable than part (c). Each of the questions in this task could be formulated as an if and only if statement but the other implication, namely that f(x) is divisible by x-r if and only if r is a root of f. The direction not presented in this task is more straightforward and so has been left out. |
| The Missing Coefficient: | The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials. |
Vetted resources students can use to learn the concepts and skills in this topic.
| Title |
Description |
| Zeroes and factorization of a general polynomial: | In this task, students are asked to show or verify four theorems related to roots, zeroes, and factors of polynomial functions. The Fundamental theorem of Arithmetic is also mentioned. This task builds on "Zeroes and factorization of a quadratic function'' parts I and II. |
| Zeroes and factorization of a non polynomial function: | For a polynomial function f, if f(0)=0 then the polynomial f(x) is divisible by x. This fact is shown and then generalized in "Zeroes of a quadratic polynomial I, II" and "Zeroes of a general polynomial.'' Here, divisibility tells us that the quotient f(x)/x will still be a nice function -- indeed, another polynomial, save for the missing point at x=0. The goal of this task is to show via a concrete example that this nice property of polynomials is not shared by all functions. The non-polynomial function F given by F(x)=|x| is a familiar function for which property does not hold: even though F(0)=0, the quotient F(x)/x behaves badly near x=0. Indeed, its graph is broken into two parts which do not connect at x=0. |
| Zeroes and factorization of a quadratic polynomial II: | This task continues "Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in "Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. In the quadratic case, an alternative argument for why there can be at most two roots can be given using the quadratic formula and this is done in the second solution below.
This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions. Students who are familiar with the quadratic formula should be encouraged to think about the first solution which extends to polynomials of higher degree where formulas for the roots are either very complex or not possible to find. |
| Zeroes and factorization of a quadratic polynomial I: | For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x-r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. The teacher should pay close attention to the logic used in the solution to part (c) where the divisibility of ax2+bx+c by x-r is obtained not by performing long division but by using the result of long division of these polynomials; namely, that said division will result in an expression of the following form:
ax2+bx+c=(x-r)l(x)+k
where l is a linear polynomial and k is a number.
This task could be used either for assessment or for instructional purposes. If it is used for assessment, parts (a) and (b) are more suitable than part (c). Each of the questions in this task could be formulated as an if and only if statement but the other implication, namely that f(x) is divisible by x-r if and only if r is a root of f. The direction not presented in this task is more straightforward and so has been left out. |
| The Missing Coefficient: | The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials. |
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
| Title |
Description |
| Zeroes and factorization of a general polynomial: | In this task, students are asked to show or verify four theorems related to roots, zeroes, and factors of polynomial functions. The Fundamental theorem of Arithmetic is also mentioned. This task builds on "Zeroes and factorization of a quadratic function'' parts I and II. |
| Zeroes and factorization of a non polynomial function: | For a polynomial function f, if f(0)=0 then the polynomial f(x) is divisible by x. This fact is shown and then generalized in "Zeroes of a quadratic polynomial I, II" and "Zeroes of a general polynomial.'' Here, divisibility tells us that the quotient f(x)/x will still be a nice function -- indeed, another polynomial, save for the missing point at x=0. The goal of this task is to show via a concrete example that this nice property of polynomials is not shared by all functions. The non-polynomial function F given by F(x)=|x| is a familiar function for which property does not hold: even though F(0)=0, the quotient F(x)/x behaves badly near x=0. Indeed, its graph is broken into two parts which do not connect at x=0. |
| Zeroes and factorization of a quadratic polynomial II: | This task continues "Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in "Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. In the quadratic case, an alternative argument for why there can be at most two roots can be given using the quadratic formula and this is done in the second solution below.
This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions. Students who are familiar with the quadratic formula should be encouraged to think about the first solution which extends to polynomials of higher degree where formulas for the roots are either very complex or not possible to find. |
| Zeroes and factorization of a quadratic polynomial I: | For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x-r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. The teacher should pay close attention to the logic used in the solution to part (c) where the divisibility of ax2+bx+c by x-r is obtained not by performing long division but by using the result of long division of these polynomials; namely, that said division will result in an expression of the following form:
ax2+bx+c=(x-r)l(x)+k
where l is a linear polynomial and k is a number.
This task could be used either for assessment or for instructional purposes. If it is used for assessment, parts (a) and (b) are more suitable than part (c). Each of the questions in this task could be formulated as an if and only if statement but the other implication, namely that f(x) is divisible by x-r if and only if r is a root of f. The direction not presented in this task is more straightforward and so has been left out. |
| The Missing Coefficient: | The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials. |
