Vocabulary
Repeated Solution- For the polynomial equation f(x) + 0, k is a repeated solution if and only if the factor x - k has an exponent greater that 1 when f(x) is factored completely.
Irrational Conjugates- A conjugate with an irrational number as one of its terms.
Complex conjugates- A conjugate that is a complex number consisting of a real and imaginary number.
Irrational Conjugates- A conjugate with an irrational number as one of its terms.
Complex conjugates- A conjugate that is a complex number consisting of a real and imaginary number.
Essential Information
The Fundamental Theorem of Algebra
Theorem: If f(x) is a polynomial of degree n where n>0, ten the equation f(x) + 0 has at least one solution in the set of complex numbers.
Corollary: 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 2 solutions, each solution repeated 3 times is counted as 3 solutions and so on.
The corollary to the fundamental theorem of algebra also implies that an nth degree polynomial function has exactly n zeros.
Theorem: If f(x) is a polynomial of degree n where n>0, ten the equation f(x) + 0 has at least one solution in the set of complex numbers.
Corollary: 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 2 solutions, each solution repeated 3 times is counted as 3 solutions and so on.
The corollary to the fundamental theorem of algebra also implies that an nth degree polynomial function has exactly n zeros.
Behavior Near Zeros
The graph of the above is shown. Only real zeros appear as x intercepts. Also the graph is tangent at the repeated x intercept x=-1, but crosses the x axis at zero x+2.
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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.
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.
Descartes' Rule of Signs
Descrates found a relationship between the coefficients of a polynomial function and the number of positive and negative zeros of the function.
Let f(x) = anxⁿ + an-₁xⁿ⁻¹ + ... + a₂x + a₁x +a₀ be a polynomial function with real coefficients.
Descrates found a relationship between the coefficients of a polynomial function and the number of positive and negative zeros of the function.
Let f(x) = anxⁿ + an-₁xⁿ⁻¹ + ... + a₂x + a₁x +a₀ be a polynomial function with real coefficients.
- The number of positive real zeros of f is equal to the number of the number of changes in sign of the coefficients of f(x) or is less than this by an even number.
- The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.
sometimes zeroes can be approximated from a graph.