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List all possible rational zeros for f(x)= 5x^5 +32x^4 -5x -32. Find all real zeros of...

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cassmith111 | (Level 2) Honors

Posted February 24, 2012 at 11:28 AM via web

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List all possible rational zeros for f(x)= 5x^5 +32x^4 -5x -32. Find all real zeros of the polynomial below and factor completely.

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted February 24, 2012 at 11:39 AM (Answer #1)

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The function f(x) = 5x^5 +32x^4 -5x -32 = 0

5x^5 +32x^4 -5x -32 = 0

=> 5x^5 + 5x^4 + 27x^4 + 27x^3 - 27x^3 - 27x^2 + 27x^2 + 27x - 32x - 32 = 0

=> 5x^4(x + 1) + 27x^3(x + 1) - 27x^2(x + 1) + 27x(x + 1) - 32(x + 1) = 0

=> (x +1 )(5x^4 + 27x^3 - 27x^2 + 27x - 32) = 0

=> (x +1 )(5x^4 - 5x^3 + 32x^3 - 32x^2 + 5x^2 - 5x + 32x - 32) = 0

=> (x +1 )(5x^3(x - 1) + 32x^2(x - 1) + 5x(x - 1) + 32(x - 1)) = 0

=> (x +1 )(x - 1)(5x^3 + 32x^2 + 5x + 32) = 0

=> (x +1 )(x - 1)(x^2(5x + 32) + 1(5x + 32)) = 0

=> (x +1 )(x - 1)(5x + 32)(x^2 + 1) = 0

The real roots are x = 1, x = -1 and x = -32/5

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