Descartes' Rule of Signs
Presentation
•
Mathematics
•
10th Grade
•
Hard
Joseph Anderson
FREE Resource
15 Slides • 36 Questions
1
Determining the number of positive, negative and imaginary roots using Descartes’ rule of signs
2
Workplan
Routines
Review
Discussion
Activity
Open- ended Question
Wrap- Up
3
Big Idea
"Understanding the relationship of the remainder theorem and factor theorem will help you understand the zeroes of the polynomial function."
4
E. U.
"How will you determine the number of positive, negative and imaginary roots using the Descartes Rule of sign?"
5
Multiple Choice
f(x) = x3 + 2x2 - 6x + 8
6
Multiple Choice
f(x) = 3x3 + 2x2 - 6x + 7
7
Multiple Choice
f(x) = 2x3 + 5x2 - 9x + 5
8
Multiple Choice
2x3 - 11x2 + 12x + 9 = 0
9
Multiple Choice
10
Multiple Choice
11
Multiple Choice
All possible rational roots are zero.
True
False
Both
None of the Above
12
Descartes' Rule of Sign
13
Descartes' rule of sign is used to determine the number of real zeros of a polynomial function.
1) The number of positive real zeros of f(x) is either equal to the number of variations in sign of f(x) or is less than that number by an even integer.
2) The number of negative real zeros of f(x) is either equal to the number of variations in sign of f(-x) or is less than that number by an even integer.
14
Descartes’ rule of signs
The number of positive real roots is at most equal to the number of sign changes.
The number of roots could be less by multiples of two. After replacing x with -x, the number of negative real roots is at most equal to the number of sign changes.
15
Steps:
1. Always write the polynomial in the standard form, i.e., descending order of the exponents of the variable, and avoid the terms with coefficient 0.
2. To find the maximum number of positive real roots, count the number of sign changes in f(x), and to find the maximum number of negative real roots, count the number of sign changes in f(-x).
3. Find the maximum number of imaginary roots by by subtracting the sum of positive and real roots from the degree of the polynomial.
Note: The number of roots of a polynomial function is equal to the degree of the polynomial.
16
Descartes' rule of Signs Chart
17
Example:
Example: f(x) = x3 - x2 + x - 1.
Determine the possible number of positive, negative and imaginary roots.
18
Example:
A polynomial function f(x) (whose terms are arranged in the descending order of the powers of variable) cannot have more positive real roots than the number of sign changes in it.
"A polynomial function f(x) in standard form cannot have more negative real roots than the number of sign changes in f(-x)."
Example: f(x) = x3 - x2 + x - 1.
19
20
21
22
Multiple Choice
f(x) = x3+7x2-5x+19
Step 1. How many variations or change of signs are there?
1
2
3
4
23
Multiple Choice
Use Descartes Rule of Signs to determine the possible number of positive roots the following polynomial equation.
5x4-9x3+ 5x2-5x-4=0
4 , 2 or none
3 or 1
4
3
24
Multiple Choice
When looking for possible positive roots, we need to look at...
f(x)
f(−x)
−f(x)
xn (highest power)
25
Fill in the Blanks
Type answer...
26
Multiple Choice
How many negative solutions does the following polynomial equation have?
6x5 - 2x4 + x2 - 7x +1=0
1
4,2 or none
3 or 1
3
27
Multiple Choice
How many possible negatives solutions does the following polynomial equation have?
2x4 - x3 - 10x2 + 3x -4=0
1
2 or none
3
3 or 1
28
Multiple Choice
How do you use Descartes' Rule of Signs to determine the number of positive real zeros of a polynomial?
Divide the polynomial by its highest degree term.
Count the number of negative real zeros of the polynomial.
Find the sum of the coefficients of the polynomial.
Count the number of sign changes in the coefficients of the polynomial.
29
Multiple Choice
How do you use Descartes' Rule of Signs to determine the number of negative real zeros of a polynomial?
Count the number of sign changes in the coefficients of the polynomial to determine the number of complex zeros.
Count the number of sign changes in the coefficients of the polynomial to determine the number of positive real zeros.
Count the number of sign changes in the coefficients of the polynomial to determine the number of negative real zeros, or less than that by an even number.
Descartes' Rule of Signs cannot be used to determine the number of negative real zeros of a polynomial.
30
Multiple Choice
What is the maximum number of positive real zeros a polynomial can have?
Number of terms in the polynomial
Constant term of the polynomial
Degree of the polynomial
Leading coefficient of the polynomial
31
Multiple Choice
What is the maximum number of negative real zeros a polynomial can have?
The number of non-real zeros
The number of complex zeros
The number of positive real zeros
The degree of the polynomial
32
Fill in the Blanks
Type answer...
33
Multiple Select
x5+3x2−4x+120 has how many total possible positive real solutions?
4
3
2
1
0
34
Multiple Select
x5+3x2−4x+120 has how many total possible negative real solutions?
4
3
2
1
5
35
Multiple Select
x5+3x2−4x+120 has how many total possible imaginary solutions?
4
3
2
1
0
36
Multiple Select
If f(x) has a maximum possible number of 5 positive real roots, what else is true?
It also has 5 negative real roots
It might also have 3 positive real roots instead
It might also have 1 positive real root instead
It might also have 3 negative real roots instead
It might also have 1 negative real root instead
37
Multiple Choice
What is the degree of a polynomial?
The degree of a polynomial is the number of terms in the polynomial.
The degree of a polynomial is the sum of the coefficients in the polynomial.
The degree of a polynomial is the highest power of the variable in the polynomial.
The degree of a polynomial is the lowest power of the variable in the polynomial.
38
Multiple Choice
How do you identify the degree of a polynomial?
The degree of a polynomial is determined by the coefficient of the highest power of the variable in the polynomial.
The degree of a polynomial is determined by the highest power of the variable in the polynomial.
The degree of a polynomial is determined by the number of terms in the polynomial.
The degree of a polynomial is determined by the lowest power of the variable in the polynomial.
39
Multiple Choice
What is Descartes' Rule of Signs?
Descartes' Rule of Signs is a mathematical rule that helps determine the number of positive and negative roots of a polynomial equation by examining the signs of its coefficients.
Descartes' Rule of Signs is a mathematical rule that helps determine the number of positive roots of a polynomial equation by examining the signs of its coefficients.
Descartes' Rule of Signs is a mathematical rule that helps determine the number of negative roots of a polynomial equation by examining the signs of its coefficients.
Descartes' Rule of Signs is a mathematical rule that helps determine the number of real roots of a polynomial equation by examining the signs of its coefficients.
40
Multiple Choice
How do you use Descartes' Rule of Signs to determine the number of positive real zeros of a polynomial?
Divide the polynomial by its highest degree term.
Count the number of negative real zeros of the polynomial.
Find the sum of the coefficients of the polynomial.
Count the number of sign changes in the coefficients of the polynomial.
41
Multiple Choice
How do you use Descartes' Rule of Signs to determine the number of negative real zeros of a polynomial?
Count the number of sign changes in the coefficients of the polynomial to determine the number of complex zeros.
Count the number of sign changes in the coefficients of the polynomial to determine the number of positive real zeros.
Count the number of sign changes in the coefficients of the polynomial to determine the number of negative real zeros, or less than that by an even number.
Descartes' Rule of Signs cannot be used to determine the number of negative real zeros of a polynomial.
42
Multiple Choice
What is the maximum number of positive real zeros a polynomial can have?
Number of terms in the polynomial
Constant term of the polynomial
Degree of the polynomial
Leading coefficient of the polynomial
43
Multiple Choice
What is the maximum number of negative real zeros a polynomial can have?
The number of non-real zeros
The number of complex zeros
The number of positive real zeros
The degree of the polynomial
44
Multiple Choice
How many solution does x³-12x+35x-6=0 have?
1
2
3
4
45
Multiple Choice
How many zeroes does the function f(x)= x⁵-3x⁴+3x³+25x+10 have?
5
4
3
2
46
Multiple Choice
How many solutions does x²+3x-3=0 have?
0
1
2
3
47
Multiple Choice
When looking for possible imaginary/complex roots, we need to look at...
f(x)
f(−x)
−f(x)
xn (highest power)
48
Multiple Choice
Use Descartes Rule of Signs to determine the possible number of positive and negative roots.
3 or 1 positive roots and 0 negative roots
4, 2, or 0 positive roots and 1 negative root
4, 2, or 0 positive roots and 0 negative roots
2 or 0 positive roots and 1 negative roots
49
Multiple Choice
Find a possible number of positive, negative and complex roots combination of the following polynomial
7𝑥5 - 3𝑥2 - 𝑥 + 9 = 0
2, 2 ,1
2,1, 2
3,1,1
2, 2,1
50
Open Ended
How will you determine the number of positive, negative and imaginary roots using the Descartes Rule of sign?
51
Wrap- Up
"Descartes' rule of signs is used to find the maximum number of positive and negative real roots. To find the maximum number of positive real roots, count the number of sign changes in f(x), and to find the maximum number of negative real roots, count the number of sign changes in f(-x)."
Determining the number of positive, negative and imaginary roots using Descartes’ rule of signs
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