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Section 5.5 – The Real Zeros of a Rational Function

Section 5.5 – The Real Zeros of a Rational Function. Remainder Theorem. If f(x) is a polynomial function and is divided by x – c, then the remainder is f(c). Example:. T he remainder after dividing f(x) by (x – 4) would be -7. Section 5.5 – The Real Zeros of a Rational Function.

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Section 5.5 – The Real Zeros of a Rational Function

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  1. Section 5.5 – The Real Zeros of a Rational Function Remainder Theorem • If f(x) is a polynomial function and is divided by x – c, then the remainder is f(c). • Example: The remainder after dividing f(x) by (x – 4) would be -7.

  2. Section 5.5 – The Real Zeros of a Rational Function • Factor Theorem • If f(x) is a polynomial function, then x – c is a factor of f(x) if and only if f(c) = 0. • Example: The remainder after dividing f(x) by (x + 3) would be 0.

  3. Section 5.5 – The Real Zeros of a Rational Function Rational Zeros Theorem (for functions of degree 1 or higher) (1) Given: (2) Each coefficient is an integer. If (in lowest terms) is a rational zero of the function, then p is a factor of and q is a factor of . • Theorem: A polynomial function of odd degree with real coefficients has at least one real zero.

  4. Section 5.5 – The Real Zeros of a Rational Function Rational Zeros Theorem Example: Find the solution(s) of the equation. Possible solutions: Try:

  5. Section 5.5 – The Real Zeros of a Rational Function Long Division Synthetic Division

  6. Section 5.5 – The Real Zeros of a Rational Function

  7. Section 5.5 – The Real Zeros of a Rational Function Example: Find the solution(s) of the equation. Possible solutions : Try: Try: Try:

  8. Section 5.5 – The Real Zeros of a Rational Function

  9. Section 5.5 – The Real Zeros of a Rational Function Intermediate Value Theorem In a polynomial function, if a < b and f(a) and f(b) are of opposite signs, then there is at least one real zero between a and b.

  10. Section 5.5 – The Real Zeros of a Rational Function Intermediate Value Theorem Do the following polynomial functions have at least one real zero in the given interval?

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