Problem 6. Let P(x) 4z6 -825 +82 -4z3 +822 -8z. (i) Using Descartes\' Rules of S

Question

Problem 6. Let P(x) 4z6 -825 +82 -4z3 +822 -8z. (i) Using Descartes' Rules of Signs determine the number of real positive roots of P() (i) Use the Rational Root Theorem to determine all possible rational roots of P() (Hint: (ii) Use the Bounds Theorem to determine an upper bound for the modulus of roots of (iv) Evaluate P(z) at possible rational roots and use the Factor Theorem to find one or (v) Given that is a root of Plz), find all roots of P) and the number of real negative roots of P(z). What is the minimum number of real roots? What is the maximum? If P(r)-Q() R(a) where Q(x) and R(a) are polynomials, then the rational roots of Q(x) and R(x) are also rational roots of P(x)) P(c). Does this eliminate any possible rational roots? Which ones? more linear factor(s) which divide P()

Explanation / Answer

P(x) = 4x^6 -8x^5 + 8x^4 - 4x^3 + 8x^2 - 8x

x ( 4x^5 -8x^4 + 8x^3 - 4x^2 + 8x - 8)

total number of positive roots are 5

since there are 5 sign changes

P(-x) = -4x^6 +8x^5 - 8x^4 +4x^3 - 8x^2 + 8x

total sign changes = 5

so total number of negative roots are 5

maximum number of real roots can be 6

minimum number of real roots = 0

ii) polynomial rational roots are +- 1, 2, 4 , 8 / +- 1, 2 , 4

possible rational roots are +1 /1 , -1 /1 , +4/2 , -4/2 , +2/1 , - 2/1 , +2/4 , -2/4 , + 4/1 , -4/1 , +8/1 , -8/1

iii) dividing entire polynomial by 4

4x^6 -8x^5 + 8x^4 - 4x^3 + 8x^2 - 8x

x^6 - 2x^5 + 2x^4 - x^3 + 2x^2 - 2x

coefficients of polynomial are

1 , -2 , 2 , -1 , 2 , -2

drop the leading coefficient and remove the minus sign

2,2,1,2,2

bound 1 = 2 +1 = 3

bound 2 = 9

smallest bound is -3 and + 3

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