FINDING ROOTS WITHOUT A CLUE

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Presentation transcript:

FINDING ROOTS WITHOUT A CLUE Rational Zero Theorem: If has integer coefficients, then every rational zero of f has the following form: p/q=factors of constant term/factors of leading coefficient term

USING THE RATIONAL ZERO THEOREM List the possible rational zeros of the function using the rational zero theorem: 1. 2. 3. 4.

Find all of the roots of: g(x) = x4 + 2x3 – 5x2 – 4x + 6 (-3 and 1 using the Rational Zero Theorem ) Find all of the roots of: g(x) = x4 + 2x3 – 5x2 – 4x + 6 ANSWER: -3, 1, ,

Find all of the roots of: g(x) = 4x3 – 16x2 + 11x + 3 (3 using the Rational Zero Theorem) Find all of the roots of: g(x) = 4x3 – 16x2 + 11x + 3 ANSWER: 3, ,

Find all of the roots of: g(x) = x3 – x2 – 11x + 3 (-3 using the Rational Root Theorem) Find all of the roots of: g(x) = x3 – x2 – 11x + 3 ANSWER: -3, ,

Conjugate Pairs If a + bi is a zero Then a – bi is a zero

Imaginary zeros don’t cross x-axis Imaginary zeros don’t cross x-axis!!!! They are imaginary you can’t see them on a graph.

Find all of the zeros. zeros: x = 0 x = 2i x = -2i Always look to see if a function will factor. This function WILL factor. zeros: x = 0 x = 2i x = -2i

x = -1, 2 from the Rational Root Theorem Find all the roots x = -1, 2 from the Rational Root Theorem f(x) = x4 – 5x3 + 7x2 + 3x - 10 -1 1 -5 7 3 -10 -1 6 -13 10 2 1 -6 13 -10 2 -8 10 1 -4 5 These are all the roots

Find all of the zeros of the function: 5 roots: 3 real 2 imag. BOUNCES! x = -2 x = 1 x = 1

x = -2 x = 1 -2 1 1 2 -12 8 x = 1 -2 4 -10 16 -8 1 1 -2 5 -8 4 1 -1 4 -4 Zeros:: 1 1 -1 4 -4 x = -2 x = 1 1 4 x = 1 x = 2i x = -2i 1 4 f(x) = x2 + 4 0 = x2 + 4 -4 = x2 -2i, 2i = x