Topicmf507e64b1e586aa0_1528449000663_0Topic

Roots of polynomial equations

Levelmf507e64b1e586aa0_1528449084556_0Level

Third

Core curriculummf507e64b1e586aa0_1528449076687_0Core curriculum

III. Equations and inequalities. The student:

6) solves equations in polynomial form of $W\left(x\right)=0$ for polynomials simplified to a factored form or the ones which can be simplified to a factored form by factoring out the common factor or using the grouping method.

Timingmf507e64b1e586aa0_1528449068082_0Timing

45 minutes

General objectivemf507e64b1e586aa0_1528449523725_0General objective

1. Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesmf507e64b1e586aa0_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Factorizing polynomials.

3. Solving polynomial equations presented in the factored form.

Learning outcomesmf507e64b1e586aa0_1528450430307_0Learning outcomes

The student:

- factorizes a polynomial.

- solves polynomial equations presented in the factored form.

Methodsmf507e64b1e586aa0_1528449534267_0Methods

1. Mind map.

2. Discussion.

Forms of workmf507e64b1e586aa0_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmf507e64b1e586aa0_1528450127855_0Introduction

The students work in groups preparing mind maps including the most important information about equations and techniques of solving them. In particular, they recollect the equivalent equations method and the ancients’ method of analysis (analisis antiquorum).

Proceduremf507e64b1e586aa0_1528446435040_0Procedure

The teacher informs the students that the aim of the class is solving polynomial equations.

The students use the mind maps and on the analogy of other types of equations define the polynomial equation.

Definitionmf507e64b1e586aa0_1527752256679_0Definition

The polynomial equation of degree $n$ is an equationequationequationthat can be transformed equivalently to the form $W\left(x\right)=0$, where $W\left(x\right)$ is a polynomial of degree $n$$(n\in N_{+})$.

The students work with their computers. They analyse the applet, paying special attention to the number of roots of the polynomial equation. They formulate a conclusion.

[Geogebra applet]

The conclusion
In the set of real numbers the polynomial of degree $n$$(n\in N_{+})$ can have no more than $n$ roots.mf507e64b1e586aa0_1527752263647_0The conclusion
In the set of real numbers the polynomial of degree $n$$(n\in N_{+})$ can have no more than $n$ roots.

The teacher informs the students that the property they have just discovered is called the fundamental theorem of algebrafundamental theorem of algebra fundamental theorem of algebra.

Using the information, the students solve the tasks.

Task

Find the number of roots of the equationroots of the equationroots of the equation.

a) $3x^2\left(-2-x\right)\left(x^2-4\right){\left(x+5\right)}^2=0$

b) $2x^2\left(x^2+4\right){\left(x-7\right)}^2\left(x^3-8\right)=0$

c) $\left(x^2+6\right){\left(2x-5\right)}^2\left(x-3\right)=0$

Task

Check which of the numbers $\left\{-\sqrt{3},-1,1\right\}$ is the root of the polynomial.

$W\left(x\right)=x^5+x^3+a{x}^2-8$

Task

Find number $a$, when you know that number 2 is the root of the polynomial.

$W\left(x\right)=x^5+x^3+a{x}^2-8$

Discussion – what is the simplest method of solving the polynomial equation of degree larger than 2?

The students agree that the simplest method (in case of polynomials that have roots) is factorizing them to the degree no greater than 2. The students work individually to solve the task using this method.

Task
Give all the roots of the polynomial $W\left(x\right)=\left(3x-1\right)\left(x^2-9\right)\left(4x+2\right)$

Task
Factorize the polynomial $W\left(x\right)=125x^4-125x^3-8x+8$.
Then, solve the equation $W\left(x\right)=0$

Task
Solve the equation.

a) $x^4+2x^3-x-2=0$

b) $8x^4+24x^3+x+3=0$

The product of a given square number and a square number which is larger than this number by 4 equals 2025. Find these numbers.mf507e64b1e586aa0_1527712094602_0The product of a given square number and a square number which is larger than this number by 4 equals 2025. Find these numbers.

Lesson summarymf507e64b1e586aa0_1528450119332_0Lesson summary

The students do the consolidation tasks.

They cooperate to recapitulate the class and formulate conclusions to be remembered:

The polynomial equationpolynomial equationpolynomial equation of degree $n$ is an equation that can be transformed equivalently to the form $W\left(x\right)=0$, where $W\left(x\right)$ is a polynomial of degree $n$ $(n\in N_{+})$.

In the set of real numbers the polynomial of degree $n$ $(n\in N_{+})$ can have no more than $n$ roots.

Selected words and expressions used in the lesson plan

equationequationequation

polynomial equationpolynomial equationpolynomial equation

roots of the equationroots of the equationroots of the equation

number of roots of the equationnumber of roots of the equationnumber of roots of the equation

short multiplication formulasshort multiplication formulasshort multiplication formulas

fundamental theorem of algebrafundamental theorem of algebrafundamental theorem of algebra