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Use the Fundamental Theorem of Algebra to find complex zeros of a polynomial function. Use the Linear Factorization Theorem to find polynomials with given zeros. Use Descartes’ Rule of Signs to determine the maximum number of possible real zeros of a polynomial function. Solve real-world applications of polynomial equations.
Proof: Since q is a polynomial, it will be of the form: q(x) = anxn + an − 1xn − 1 +... + a1x + c. If x = cm, then notice that cm will be inside every term of the polynomial, and since the final term is c, c can be factored out of every term. Hence, q (cm) is a multiple of c for all m ∈ Z.
We say that a is a zero of the polynomial if and only if p(a) = 0. The definition also holds if the coefficients are complex, but that’s a topic for a more advanced course. For example, −5 is a zero of the polynomial p(x) = x2 + 3x − 10 because. p( − 5) = ( − 5)2 + 3( − 5) − 10 = 25 − 15 − 10 = 0.
3 Οκτ 2022 · In Section 3.2, we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section presents results which will help us determine good candidates to test using synthetic division.
16 Αυγ 2023 · In Section 3.2, we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section presents results which will help us determine good candidates to test using synthetic division.
3 Οκτ 2022 · Suppose f is a polynomial function with complex number coefficients. If the degree of f is n and n ≥ 1, then f has exactly n complex zeros, counting multiplicity. If z1, z2, …, zk are the distinct zeros of f, with multiplicities m1, m2, …, mk, respectively, then f(x) = a(x − z1)m1(x − z2)m2⋯(x − zk)mk.
30 Ιαν 2024 · To find the remaining two intercepts, we can use the quadratic equation, setting \(4x^{2} -12x+4=0\). First, we might pull out the common factor, \(4\left(x^{2} -3x+1\right)=0\). \[x=\dfrac{3\pm \sqrt{(-3)^{2} -4(1)(1)} }{2(1)} =\dfrac{3\pm \sqrt{5} }{2} \approx 2.618,\; \; 0.382\nonumber \]
