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Find a polynomial f(x) of degree 3 with real coefficients and the following zeros. 1,3-2i

Find a polynomial f(x) f(x) of degree 33 with real coefficients and the following zeros.\newline1,3−2i 1,3-2 i

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Fundamental Theorem of Algebraright chevron icon

Full solution

Q. Find a polynomial f(x) f(x) of degree 33 with real coefficients and the following zeros.\newline1,3−2i 1,3-2 i
  1. Identify Real Zero Factor: Identify the real zero and its corresponding factor.\newlineFactor for the zero 11: (x−1)(x - 1).
  2. Recognize Complex Zeros: Recognize that complex zeros occur in conjugate pairs for polynomials with real coefficients.\newlineConjugate of 3−2i3-2i is 3+2i3+2i.
  3. Write Complex Zero Factors: Write the factors corresponding to the complex zeros.\newlineFactors for zeros 3−2i3-2i and 3+2i3+2i: (x−(3−2i))(x - (3-2i)) and (x−(3+2i))(x - (3+2i)).
  4. Expand Complex Zero Factors: Expand the factors of the complex zeros.\newline(x−(3−2i))(x−(3+2i))=(x−3+2i)(x−3−2i)=x2−6x+13(x - (3-2i))(x - (3+2i)) = (x - 3 + 2i)(x - 3 - 2i) = x^2 - 6x + 13.
  5. Multiply All Factors: Multiply all factors to get the polynomial.\newline(x−1)(x2−6x+13)=x3−6x2+13x−x2+6x−13=x3−7x2+19x−13(x - 1)(x^2 - 6x + 13) = x^3 - 6x^2 + 13x - x^2 + 6x - 13 = x^3 - 7x^2 + 19x - 13.

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