how to solve Descartes Rule of Signs and Facotr Theorem

Question

 

How many positive real roots and negative roots does this problems have,

 

 

1.  f(a) = a^5 - 4a^2 - 7

 

 

 

2.  f(x) = 3x^3 + 9x^2 + 8x

 

 

how do I use the factor theorem to determine whether (x-3) is a factor of f(x) = x^4 + 12x^3 + 6x + 27

 

thanks

Answer 1

Descartes Rule of Signs says:

 

a) The maximum number of positive roots that a polynomial can have is equal to the number of sign changes (+ to - and - to +) between the terms of the polynomial f(x)

 

b) The maximum number of negative roots that a polynomial can have is equal to the number of sign changes (+ to - and - to +) between the terms of the polynomial f(-x).

 

f(a) = a⁵-4a²-7  --> One sign change (between +a⁵ and -4a²), so there is a maximum of 1 positive root.

f(-a) = (-a)⁵-4(-a)²-7 = -a⁵-4a²-7 -->  No sign changes, so there are no negative roots

 

f(x) = 3x³+9x²+8x  --> No sign changes, so no positive roots

f(-x) = 3(-x)³+9(-x)²+8(-x) = -3x³+9x²-8x --> Two sign changes, so max of 2 negative roots

 

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How do I use the factor theorem to determine whether (x-3) is a factor of f(x) = x^4 + 12x^3 + 6x + 27

 

Divide f(x) by (x-3) using regular or synthetic polynomial division.  If there is no remainder, then (x-3) is a factor of f(x).