𝑇𝐻𝐸𝑀𝐸 1 ∶ 𝑃𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙𝑠
2013
Objectives :
Production :
Organisation :
-
Vocabulary
To introduce the historical background of polynomials
To learn the english vocabulary about polynomial
To learn english methods to find roots of a polynomial
To approch historical application (Fibonacci, Law of fall)
A set of video clips
A verbal presentation of the different methods to find zeroes
A verbal exercise
A biography of Galileo, Fibonacci
4 séances de 1h
Terms coefficients and exponents in a polynomial
Listen to the youtube document. While listening, find key-words and give a translation of each in the table :
https://www.khanacademy.org/math/algebra/polynomials/polynomial_basics/v/terms-coefficients-and-exponents-in-apolynomial
Match words in the table with element of the function below :
Termes
Coefficients
𝑓(𝑥 ) = 𝑎𝑥² + 𝑏𝑥 + 𝑐
Exposant
Polynôme
Activity 2
Solving for 𝒙 a quadratic
1.
What is a quadratic ?
…………………………………………………………………………………………………………………………………………………….
2.
To find zeroes of a quadratic, just study these three methods :
a. The ac-method
http://www.youtube.com/watch?v=AYkaCZUT4O4
b.
By completing square
https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/completing-the-square
c.
The box method
http://www.youtube.com/watch?v=UXDOMz2BGWI
3.
Solve this problem using any method :
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Section européenne | HOUPERT Nicolas
2013
4. Use the French method to solve the same problem. Draw a conclusion.
5. Given these three polynomials, use each method to find zeroes of each :
𝑃(𝑥) = 𝑥 2 − 12𝑥 + 11
𝑄(𝑥) = 3𝑥² + 15𝑥 + 12
𝑅(𝑥) = −2𝑥² − 3𝑥 + 5
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2013
Activity 3
The most famous quadratic : The Fibonacci’s polynomial
Evariste Galois studied all along his « short » life a set of fractions called « continued
fractions » :
𝑥=𝑎+
Who am I ? …………………………………..
1
𝑏+
1
𝑎+
1
1
𝑏+
1
𝑎+ 1
𝑏+
…
………………………………………………………
which is also written as 𝑥 = [𝑎, 𝑏, 𝑎, 𝑏, … ] , where 𝑎, 𝑏 are two strictly positive
integers.
Let’s study the fraction where 𝑎 = 𝑏 = 1.
1. Write 𝑥.
𝑥=
2. Show that 𝑥 is a zero of the polynomial 𝑥² − 𝑥 − 1.
3. Solve for 𝑥 the equation : 𝑥² − 𝑥 − 1 = 0.
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4. Draw a conclusion to find the correct value of 𝑥.
Let’s study now the general fraction.
5. Prove that 𝑥 is a zero of the polynomial 𝑏𝑥² − 𝑎𝑏𝑥 − 𝑎.
6. Solve for 𝑥 the equation : 𝑏𝑥² − 𝑎𝑏𝑥 − 𝑎 = 0.
7. Show that 𝑥 =
𝑎𝑏+√𝑎²𝑏²+4𝑎𝑏
.
2𝑏
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8. Give the continued fraction of √2.
Activity 4
Law of falls
The work of Galileo Galilei marks the beginning of a new tradition : the full-on application of mathematics to
science. Galileo was one of the first thinkers to believe that the laws of nature are written in the language of
mathematics.
“the distance a body falls under gravity is proportional to the square of the time it takes to fall”, he said.
In simple terms, this is a mathematical formula that relates how far a body falls as a function of the time falling.
1.
What is the formula of this law of fall ? (assuming 𝑥 to be the distance of the fall, and 𝑡 the time
elapsed). Make a drawing to illustrate this statement.
Law of free fall
Who am I ?
…………………………………………………………………
………………………………………………………………..
2.
What could be the constant ?
……………………………………………………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………………………………………………
Section européenne | HOUPERT Nicolas
2013
3.
Thanks to the table, sketch the graph of the function 𝑥 → 𝑥(𝑡).
4.
What is the shape of this graph ?
………………………………………………………………………………………………………………………………………………………………
5.
𝑡
Fill in the table with data :
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
𝑥(𝑡)
𝑥(𝑡)
𝑡²
6.
What must be the constant ?
……………………………………………………………………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………………………………………………………………….
Activity 5
Interesting Polynomial
Coefficient Problem
Given a third degree polynomial, try to
find its coefficients given 2 roots and the
y-intercept.
First, on this example, locate roots and yintercept of this polynomial :
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Give a definition of both :

Definition 1 : Root of a polynomial

Definition 2 : y-intercept of a polynomial
 Now, start to listen to the youtube document and fill in the blanks.
https://www.khanacademy.org/math/algebra/polynomials/polynomial_basics/v/interesting-polynomialcoefficient-problem
In this activity, roots are A(…….. , ……..) and B(…….. , ……..). The y-intercept is P(…….. , ……..).
Give a synonym of root : ………………………………………………………………………………………………………
What is the green point on the axis ? What are the coordinates of this point ?
………………………………………………………………………………………………………………………………………………
How can you rewrite the polynomial (𝑥) = 𝑎𝑥 3 + 𝑏𝑥² + 𝑐𝑥 + 𝑑 ?
………………………………………………………………………………………………………………………………………………
What is the meaning of +𝑏 + 𝑐 + 𝑑 ?
………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………
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Section européenne | HOUPERT Nicolas