Since the polynomial has rational coefficients, complex roots occur in conjugate pairs, so, since 4 + i is a root, 4 - i must be a root as well. By the Factor Theorem, if c is a root of f(x), then x - c is a factor of f(x). Therefore, the polynomial with smallest possible degree and with leading coefficient 1 would beDemonstrates the steps involved in solving a general polynomial, including how to use the Rational Roots Test and synthetic division. The general technique for solving bigger-than-quadratic polynomials is pretty straightforward, but the process can be time-consuming.

34. Determine the zeros and the end behavior of f(x) = −x(x − 1)(x − 3)2. Then sketch a graph of the function. Zeros: _____ End behavior: _____ _____ 35. Determine the sign of the leading coefficient and the least possible degree of the polynomial for the given graph. 36. For the following function, find the intervals where the function ... Approximate the real zeros of a polynomial function using the Intermediate Value Theorem. Approximate the real zeros of a polynomial using a graphing utility. Prior to these sections the students should know how to evaluate a polynomial at a given point, factor basic polynomials, and find zeros of quadratics and factorable third degree polynomials.

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