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3.7: Zeros of Polynomial Functions
https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
4.4: Zeros of Polynomial Functions
https://math.libretexts.org/Courses/Fresno_City_College/Math_3A%3A_College_Algebra_-_Fresno_City_College/04%3A_Polynomial_and_Rational_Functions/4.04%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
3.6: Zeros of Polynomial Functions
https://math.libretexts.org/Courses/Hartnell_College/MATH_25%3A_PreCalculus_(Abramson_OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.06%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
3.5: Zeros of Polynomials
https://math.libretexts.org/Courses/Highline_College/MATH_141%3A_Precalculus_I_(2nd_Edition)/03%3A_Polynomial_and_Rational_Functions/3.05%3A_Zeros_of_Polynomials
To solve this problem, we will need a good understanding of the relationship between the $x$ -intercepts of the graph of a function and the zeros of a function, the Factor Theorem, the role of multip...To solve this problem, we will need a good understanding of the relationship between the $x$ -intercepts of the graph of a function and the zeros of a function, the Factor Theorem, the role of multiplicity, complex conjugates, the Complex Factorization Theorem, and end behavior of polynomial functions. (In short, you'll need most of the major concepts of this chapter.) Since the graph of $p$ touches the $x$ -axis at $\left(\frac{1}{3}, 0\right)$ , we know $x=\frac{1}{3}$ is a zero of eve…
4.7: Zeros of Polynomial Functions
https://math.libretexts.org/Workbench/1250_Draft_4/04%3A_Polynomial_Functions/4.07%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
1.3.7: Zeros of Polynomial Functions
https://math.libretexts.org/Workbench/Book-_Precalculus_I_for_Highline_College_w/Rational_Inequalities_and_Equations_of_Circles/1.03%3A_Polynomial_and_Rational_Functions/1.3.07%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
6.4: The Quadratic Formula
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/06%3A_Complex_Numbers/6.04%3A_The_Quadratic_Formula
When working with real numbers, we cannot solve the quadratic formula if $b^{2}-4ac<0.$ However, complex numbers allow us to find square roots of negative numbers, and the quadratic formula remains ...When working with real numbers, we cannot solve the quadratic formula if $b^{2}-4ac<0.$ However, complex numbers allow us to find square roots of negative numbers, and the quadratic formula remains valid for finding roots of the corresponding quadratic equation.
3.6: Zeros of Polynomial Functions
https://math.libretexts.org/Courses/Quinebaug_Valley_Community_College/MAT186%3A_Pre-calculus_-_Walsh/03%3A_Polynomial_and_Rational_Functions/3.06%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
3.6: Zeros of Polynomial Functions
https://math.libretexts.org/Courses/Highline_College/Math_141%3A_Precalculus_I_(old_edition)/03%3A_Polynomial_and_Rational_Functions/3.06%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .
23.3: Applications
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/23%3A_Galois_Theory/23.03%3A_Applications
Let $\alpha$ be a zero of $x^n - a\text{.}$ Since $\alpha$ and $\omega \alpha$ are both in the splitting field of $x^n - a\text{,}$ $\omega = (\omega \alpha)/ \alpha$ is also in \(E\text{....Let $\alpha$ be a zero of $x^n - a\text{.}$ Since $\alpha$ and $\omega \alpha$ are both in the splitting field of $x^n - a\text{,}$ $\omega = (\omega \alpha)/ \alpha$ is also in $E\text{.}$ Let $K = F( \omega)\text{.}$ Then $F \subset K \subset E\text{.}$ Since $K$ is the splitting field of $x^n - 1\text{,}$ $K$ is a normal extension of $F\text{.}$ Therefore, any automorphism $\sigma$ in $G(F( \omega)/ F)$ is determined by $\sigma( \omega)\text{.}$ It must be the…
3.6: Zeros of Polynomial Functions
https://math.libretexts.org/Courses/Truckee_Meadows_Community_College/TMCC%3A_Precalculus_I_and_II/Under_Construction_test2_03%3A_Polynomial_and_Rational_Functions/Under_Construction_test2_03%3A_Polynomial_and_Rational_Functions_3.6%3A_Zeros_of_Polynomial_Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainde...In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$ , the remainder may be found quickly by evaluating the polynomial function at $k$ , that is, $f(k)$ .

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