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SWBAT use Descartes’s Rule of Signs to find zeros of polynomials and write polynomial
functions given the zeros (Section 3-5)
Warm up
List all the possible rational zeros of f, and determine all real zeros of f.
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Another Test for Zeros of Polynomials
Theorem: Descartes’ Rules of Signs
Let f(x) = anxn + an-1xn-1 + …..+a1x + a0 be a polynomial with real coefficients and a0 ≠ 0.
a) The number of positive real zeros of f is either equal to the number of variations in sign
of f(x) or less than that number by an even integer.
b) The number of negative real zeros of f is either equal to the numbers of variations in
sign of f(-x) or less than that number by an even integer.
** A variation in sign means that two consecutive coefficients have opposite signs.
The rule provides information about the number and location of the real zeros of a polynomial function.
Example 1) Using Descartes’s rule of signs describe the possible real zeros of
a) g(x) = 3x 3 – 5x2 + 6x – 4
b) f(x) = 5x 5 +10x
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c) g(x) = 2x 3 – 3x2 – 3
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Example 2) Find the real and complex zeros of f ( x)  6 x 3  4 x 2  3x  2
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Extra Practice
Using Descartes’s rule of signs describe the possible real zeros of
1) f ( x)  3x 3  2 x 2  x  3
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2) g ( x)  2 x  3x  3
3) Factor completely and find all the real and complex zeros.
P(x) = x 3 + x2 – 4x – 4
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4) ) Factor completely and find all the real and complex zeros.
5) P(x) = x4 - 5x 3 + 3x2 + x
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6) ) P(x) = 3x 6 – 4x4 + 3x3 + 2x2 – x - 3
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