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Finding Real Zeros

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1. P(x)= 2x^4 + 15x^3 + 17x^2 + 3x -1
Find all real zeros.

2. P(x)= 8x^3 + 10x^2 - 39x + 9; a=-3,b=2
Show that the given values for a and b are lower and upper bounds for the real zeros of the polynomial.

3. P(x)= x^3 - 3x^2 + 4
Find integers that are upper and lower bounds for the real zeros of the polynomial.

4. P(x)= 3x^3 - x^2 - 6x + 12
Show that the polynomial does not have any rational zeros.

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1. P(x) = 2x4 + 15x3 + 17x2 + 3x - 1. Find all real zeros.
By observation, x = -1 is a zero. Therefore, x + 1 is a factor of P(x).
Therefore, P(x) = 2x4 + 15x3 + 17x2 + 3x - 1
= 2x3(x + 1) + 13x2(x + 1) + 4x(x + 1) - (x + 1)
= (x + 1)(2x3 + 13x2 + 4x - 1)
By observation, x = -0.5 is a solution of (2x3 + 13x2 + 4x - 1), therefore, (2x + 1) is a factor of (2x3 + 13x2 + 4x - 1)
Therefore, P(x) = (x + 1)(2x3 + 13x2 + 4x - 1)
= (x + 1)[(x2(2x + 1) + 6x(2x + 1) - (2x + 1)]
= (2x + 1)(x + 1)(x2 + 6x - 1)
Therefore, the zero are x = -0.5, -1 and ...

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