Section 3.4 - Zeros of Polynomials.pdf - Section 3.4 – Zeros of Polynomials Objectives Students will be able to apply the rational zero theorem and

# Section 3.4 - Zeros of Polynomials.pdf - Section 3.4 –...

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Section 3.4 Zeros of Polynomials Objectives: Students will be able to apply the rational zero theorem and apply the fundamental theorem of algebra. I. Apply the Rational Zero Theorem Rational Zero Theorem If 𝑓(𝑥) = ? 𝑛 𝑥 𝑛 + ? 𝑛−1 𝑥 𝑛−1 + ? 𝑛−2 𝑥 𝑛−2 + ⋯ + ? 1 𝑥 + ? 0 has integer coefficients and ? 𝑛 ≠ 0 , and if ? ? (written in lowest terms) is a rational zero of 𝑓 , then ? is a factor of the constant term ? 0 and ? is a factor of the leading coefficient ? 𝑛 . If a rational zero exists for a polynomial, then it must be of the form: ? ? = 𝐅𝐚????? ?? 𝒂 𝟎 (?????𝐚?? ????) 𝐅𝐚????? ?? 𝒂 𝒏 (??𝐚?𝐢?? ?????𝐢?𝐢???) Ex. 1 List all possible rational zeros of 𝑓(𝑥) = 4𝑥 3 − 13𝑥 2 − 32𝑥 − 15 . Ex. 2 Find the zeros and their multiplicities. 𝑓(𝑥) = 2𝑥 4 + 𝑥 3 − 5𝑥 2 − 𝑥 + 3 . Ex. 3 Find the zeros and their multiplicities. 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 14𝑥 − 5 .
II. Apply the Fundamental Theorem of Algebra Fundamental Theorem of Algebra If 𝑓(𝑥) is a polynomial of degree

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