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5.5 Polynomial Division

5.5 Polynomial Division. Evaluate a polynomial Direct Substitution Synthetic Substitution Polynomial Division Long Division Synthetic Division Remainder Theorem Factor Theorem. Remainder Theorem. If a polynomial is divided by x-k, then the remainder r = f(k). Factor Theorem.

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5.5 Polynomial Division

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  1. 5.5 Polynomial Division • Evaluate a polynomial • Direct Substitution • Synthetic Substitution • Polynomial Division • Long Division • Synthetic Division • Remainder Theorem • Factor Theorem

  2. Remainder Theorem • If a polynomial is divided by x-k, then the remainder r = f(k).

  3. Factor Theorem • Polynomial f(x) has a factor x – k, if and only if f(k) = 0. • In other words… • If r is 0, then x-k is a factor of f(x) • If x-k is a factor, then k is a zero (root) of f(x).

  4. The following statements mean the same thing… • 3 is a zero of f(x). • f(3) = 0 • 3 is an x‑intercept of the graph of f(x). • (x ‑ 3) is a factor of f(x). • f(x) divided by (x ‑ 3) has a remainder of 0. • 3 is a root of f(x)

  5. p. 367 #41 • The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled by P = -x3 + 4x2 + x where x is the # of t-shirts produced (in millions). Currently the company produces 4 million t-shirts and makes a profit of 4 million. What lesser number of t-shirts could the company produce and still make the same profit?

  6. Rational Zero (Root) Theorem • If f(x) = anxn + …+ a1x + ao has integer coefficients, then every rational zero of f(x) has the following form:

  7. Fundamental Theorem of Algebra • Every polynomial of degree n where n>0, has at least one zero, where a zero may be a complex number (a + bi). • Corollary: • Every polynomial of degree n where n>0, has exactly n zeros, including multiplicities.

  8. 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. 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, then a –√b is also a rational zero of f.

  9. If f(x) is a polynomial if degree n where n>0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three time is counted as three solutions…

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