Chapter 2 – Polynomial and Rational Functions 2.5 – The Fundamental Theorem of Algebra.

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Presentation transcript:

Chapter 2 – Polynomial and Rational Functions 2.5 – The Fundamental Theorem of Algebra

Factoring Review Two quick factoring notes: 1. Ex: Factor f(x) = x 4 + 4x ◦Because each exponent of x is a multiple of a quadratic, factor it like you would factor a quadratic! ◦If f(x) = x 2 + 4x + 3, we’d factor it to… ◦f(x) = (x + 3)(x + 1) ◦But it’s a multiple of x 2, so we really have… ◦f(x) = (x 2 + 3)(x 2 + 1) 2. Any quadratic of the form f(x) = ax 2 + c, where c>0, is not factorable OVER THE REALS, but if c<0, it is factorable over the reals. o Ex: f(x) = x isn’t factorable, but x 2 – 7 is.

Fundamental Thm. of Algebra: If f(x) is a polynomial of degree n, then f has at least one zero in the complex number system. ◦The zeros may be real or complex! Linear Factorization Theorem: If f(x) is a polynomial of degree n, then f has precisely n linear factors… ◦c 1, c 2,…, c n are complex numbers

Complex Zeros Ex: Find all complex zeros of f(x) = x 3 + 4x. ◦Factor! ◦We could use synthetic division, but let’s factor out an x first. ◦f(x) = x(x 2 + 4) ◦We know 0 is a solution from the factored x ◦Since (x 2 + 4) isn’t factorable, use quadratic formula to obtain the roots… ◦So the zeros are at x = 0, -2i, and 2i.

Complex Zeros Conjugate pair rule = If a + bi is a zero of a polynomial f, then so is a – bi. Ex: Find a 4 th -degree polynomial with -1, 2, and 3i as factors. ◦Since 3i is a factor, so is -3i ◦Write as a linear factorization and multiply it out! ◦f(x) = (x + 1)(x – 2)(x + 3i)(x – 3i) ◦f(x) = (x 2 – x – 2)(x 2 + 9) ◦f(x) = x 4 + 9x 2 – x 3 – 9x – 2x 2 – 18 ◦f(x) = x 4 – x 3 + 7x 2 – 9x – 18

Find all complex zeros of: f(x) = x 4 – 3x 3 – 9x 2 – 3x – , -1, 2, , 1, 5i, -5i 3. 2, -5, i, i 4. -2, 5, i, -i

Find a polynomial of degree 4 with zeros at x = ½, 0, and -2i. 1. f(x) = 2x 4 – 5x 3 + 8x 2 2. f(x) = 2x 4 – x 3 + 8x 2 – 4x 3. f(x) = 2x 4 + 7x 2 – 4 4. f(x) = 2x 4 – x 2 + 8x – 4

Factoring Review Completely factor the following polynomials: x 4 – 16 ◦(x – 2)(x + 2)(x 2 +4)  Irreducible over the reals ◦(x – 2)(x + 2)(x + 2i)(x – 2i)  -----FINAL ANSWER x 4 – x 2 – 20 ◦(x 2 – 5)(x 2 + 4)  Irreducible over the rationals ◦  Irreducible over the reals ◦  ----FINAL ANSWER x 5 + x 3 + 2x 2 – 12x + 8 ◦Use synthetic division (with the help of your calculator)! ◦(x – 1) 2 (x + 2)(x 2 + 4)  Irreducible over the reals ◦(x – 1) 2 (x + 2)(x + 2i)(x – 2i)  FINAL ANSWER