# Finding Real Zeros

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.

See attached file for full problem description.

Â© BrainMass Inc. brainmass.com March 4, 2021, 8:05 pm ad1c9bdddfhttps://brainmass.com/math/basic-algebra/finding-real-zeros-143388

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

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 ...

#### Solution Summary

Real zeros of polynomials are found. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.