Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra
Learning Objectives for CH 2.7
Apply the Fundamental Theorem of Algebra If you factor some polynomial equations, such as x3 + 2x2 7x + 4 = 0, you get:
Apply the Fundamental Theorem of Algebra • Fundamental Theorem of Algebra • Repeated solution • Conjugates • Rule of Signs
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra However, you get a repeated solution, due to the exponent.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has at least one solution in the set of complex numbers.
Note: A complex solution does not necessarily require an imaginary component, because complex numbers are comprised of real and imaginary numbers.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra Corollary to The Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
How many solutions does do the following equations have?
If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has exactly n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three times is counted as three solutions, and so on. This implies that an nth degree polynomial has n zeros.
Jul 95:57 PM
Jul 95:57 PM
1
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra
Compare the next two graphs. Notice what happens when a factor is raised to an even power and when the factor is raised to an odd power.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
When the factor ( x k ) is raised to an odd exponent, the graph crosses the xaxis at x = k.
When the factor ( x k ) is raised to an even exponent, the graph is tangent to the xaxis at x = k.
Jul 95:57 PM
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra Complex Conjugates Theorem If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a bi is also a zero of f.
Irrational Conjugates Theorem Suppose f is a polynomial function with rational coefficients, and a and b are rational numbers such that √b is irrational. If a + √b is a zero of f then a √b is also a zero of f.
Jul 95:57 PM
2
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra Why are the Complex Conjugates Theorem and Irrational Conjugates Theorem true? Because of the QUADRATIC FORMULA!
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra It all comes from the ± symbol in the quadratic formula.
The ± symbol always gives you two answers, so when you get one complex answer, there must be a second complex answer that goes along with it.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra For the function f(x) = x3 2x + 5, there is one real solution that we can identify visually, and the degree of the function tells us that there are 3 solutions, so the remaining two must be complex, because complex solutions always come in pairs.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Write the function of least degree that has a leading coefficient of 1 with 2 and 3√2 as zeros.
Jul 95:57 PM
This is a cool step!
Jul 95:57 PM
3
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra
Descartes' Rule of Signs Let be a polynomial function with real coefficients.
How many positive real zeros, negative real zeros and imaginary zeros does the function possess?
• The number of positive real zeros of is equal to the number of changes in the sign of the coefficients of or is less than this by an even number. • The number of negative real zeros of is equal to the number of changes in the sign of the coefficients of or is less than this by an even number.
Jul 95:57 PM
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra
first sign change
first sign change
second sign change
second sign change
third sign change
Positive Negative Imaginary Real Real Zeros Zeros Zeros 2 3 0 2 1 2 0 3 2 0 1 4
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Total Zeros 5 5 5 5
Jul 183:54 PM
Apply the Fundamental Theorem of Algebra
Learning Objectives for CH 2.7 Apply the Fundamental Theorem of Algebra one real zero, and four imaginary zeros
• Fundamental Theorem of Algebra • Repeated solution • Conjugates • Rule of Signs
Jul 183:58 PM
Jul 95:57 PM
4
Learning Objectives for CH 2.7
Apply the Fundamental Theorem of Algebra If you factor some polynomial equations, such as x3 + 2x2 7x + 4 = 0, you get:
Apply the Fundamental Theorem of Algebra • Fundamental Theorem of Algebra • Repeated solution • Conjugates • Rule of Signs
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra However, you get a repeated solution, due to the exponent.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has at least one solution in the set of complex numbers.
Note: A complex solution does not necessarily require an imaginary component, because complex numbers are comprised of real and imaginary numbers.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra Corollary to The Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
How many solutions does do the following equations have?
If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has exactly n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three times is counted as three solutions, and so on. This implies that an nth degree polynomial has n zeros.
Jul 95:57 PM
Jul 95:57 PM
1
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra
Compare the next two graphs. Notice what happens when a factor is raised to an even power and when the factor is raised to an odd power.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
When the factor ( x k ) is raised to an odd exponent, the graph crosses the xaxis at x = k.
When the factor ( x k ) is raised to an even exponent, the graph is tangent to the xaxis at x = k.
Jul 95:57 PM
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra Complex Conjugates Theorem If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a bi is also a zero of f.
Irrational Conjugates Theorem Suppose f is a polynomial function with rational coefficients, and a and b are rational numbers such that √b is irrational. If a + √b is a zero of f then a √b is also a zero of f.
Jul 95:57 PM
2
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra Why are the Complex Conjugates Theorem and Irrational Conjugates Theorem true? Because of the QUADRATIC FORMULA!
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra It all comes from the ± symbol in the quadratic formula.
The ± symbol always gives you two answers, so when you get one complex answer, there must be a second complex answer that goes along with it.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra For the function f(x) = x3 2x + 5, there is one real solution that we can identify visually, and the degree of the function tells us that there are 3 solutions, so the remaining two must be complex, because complex solutions always come in pairs.
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Write the function of least degree that has a leading coefficient of 1 with 2 and 3√2 as zeros.
Jul 95:57 PM
This is a cool step!
Jul 95:57 PM
3
Algebra 2 Ch 2.7 Apply the Fundamental Theorems of Algebra for PDF.notebook October 17, 2014 Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra
Descartes' Rule of Signs Let be a polynomial function with real coefficients.
How many positive real zeros, negative real zeros and imaginary zeros does the function possess?
• The number of positive real zeros of is equal to the number of changes in the sign of the coefficients of or is less than this by an even number. • The number of negative real zeros of is equal to the number of changes in the sign of the coefficients of or is less than this by an even number.
Jul 95:57 PM
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Apply the Fundamental Theorem of Algebra
first sign change
first sign change
second sign change
second sign change
third sign change
Positive Negative Imaginary Real Real Zeros Zeros Zeros 2 3 0 2 1 2 0 3 2 0 1 4
Jul 95:57 PM
Apply the Fundamental Theorem of Algebra
Total Zeros 5 5 5 5
Jul 183:54 PM
Apply the Fundamental Theorem of Algebra
Learning Objectives for CH 2.7 Apply the Fundamental Theorem of Algebra one real zero, and four imaginary zeros
• Fundamental Theorem of Algebra • Repeated solution • Conjugates • Rule of Signs
Jul 183:58 PM
Jul 95:57 PM
4
