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- 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_PolynomialsTo 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…
- https://math.libretexts.org/Workbench/1250_Draft_4/04%3A_Polynomial_Functions/4.07%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.04%3A_Polynomial_and_Rational_Functions/1.4.05%3A_Zeros_of_PolynomialsTo 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…
- https://math.libretexts.org/Workbench/Algebra_and_Trigonometry_2e_(OpenStax)/05%3A_Polynomial_and_Rational_Functions/5.06%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Workbench/College_Algebra_2e_(OpenStax)/05%3A_Polynomial_and_Rational_Functions/5.06%3A_Zeros_of_Polynomial_FunctionsThe Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial The...The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial The Rational Zero Theorem states that, if the polynomial \(f(x)=a_nx^n+a_{n−1}x^{n−1}+...+a_1x+a_0\) has integer coefficients, then every rational zero of \(f(x)\) has the form \(\frac{p}{q}\) where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).
- https://math.libretexts.org/Courses/Mission_College/Math_1X%3A_College_Algebra_w__Support_(Sklar)/03%3A_Polynomial_and_Rational_Functions/3.04%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Courses/College_of_the_Desert/Math_10%3A_College_Algebra/05%3A_Polynomial_and_Rational_Functions/5.05%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_1e_(OpenStax)/05%3A_Polynomial_and_Rational_Functions/506%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Courses/Coastline_College/Math_C115%3A_College_Algebra_(Tran)/05%3A_Polynomial_and_Rational_Functions/5.06%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Courses/Chabot_College/Chabot_College_College_Algebra_for_BSTEM/05%3A_Polynomial_and_Rational_Functions/5.05%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
- https://math.libretexts.org/Courses/Palo_Alto_College/College_Algebra/03%3A_Polynomial_and_Rational_Functions/3.06%3A_Zeros_of_Polynomial_FunctionsIn 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, \(f(k)\).
