Apply the Fundamental Theorem of Algebra

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Notes 6.6 Fundamental Theorem of Algebra

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Presentation transcript:

Apply the Fundamental Theorem of Algebra Section 5-7 Apply the Fundamental Theorem of Algebra

Example 1 How many solutions does each equation have? a.) x4 + 8x2 – 5x + 2 b.) x3 + x2 – 3x – 3 Four Three = 5g3(g + 4)(g – 4)

Example 2 Find all real zeros of f(x) = x5 – 4x3 + x2 – 4 Step 1: List possible rational zeros. ± 1, ± 2, ± 4 ±1 Step 2: Test the zeros using synthetic division. Test x = -1 -1 1 0 – 4 1 0 –4 -1 1 3 -4 4 1 -1 -3 4 -4 x4 – x3 – 3x2 + 4x – 4

Example 2 Find all real zeros of f(x) = x4 – x3 – 3x2 + 4x – 4 Step 3: List possible rational zeros. ± 1, ± 2, ± 4 ±1 Step 4: Test the zeros using synthetic division. Test x = -2 -2 1 -1 – 3 4 –4 -2 6 -6 4 1 -3 3 -2 x3 – 3x2 + 3x – 2 = 5g3(g + 4)(g – 4)

Example 2 - continued f(x) = x3 – 3x2 + 3x – 2 Step 5: Test the zeros using synthetic division. Test x = 2 2 1 – 3 3 -2 2 2 -2 1 - 1 1 x2 – x + 1

Example 2 - continued Step 6: Find the remaining zeros by solving x2 – x + 1 = 0. Use the quadratic formula: x = -b ± √b2 – 4ac 2a x = 1 ± i√3 2 x = 1 ± √(-1)2 – 4(1)(1) 2(1) x = 1 ± √1 – (4) 2 x = 1 ± √-3 2 The real zeros of f are -1, 2, -2, 1 + i√3, 1 – i√3 2 2 = 5g3(g + 4)(g – 4)

Homework Section 5-7 Pages 383 – 384 3 – 6, 9, 10 13 – 16

The Fundamental Theorem of Algebra Theorem: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution in the set of complex numbers. Corollary: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions and so on.