Find all rational zeros of the polynomial, and then find the irrational zeros, If any, Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes's Rule of signs, the Quadratic Formula, or other factoring techniques, (Enter your answers as comma-separated lists, Enter all answers using the appropriate multiplicities. I answer does not exist, enter DNL,) \[ P(x)=2 x^{4}+11 x^{3}-8 x^{2}-11 x+6 \] rational zeros \( x=\square \) Irrational zeros \( x=\square \)

We want to find all the zeros of   P(x) = 2x⁴ + 11x³ − 8x² − 11x + 6. Step 1. Use the Rational Zeros Theorem. The possible rational zeros are of the form ±(factor of constant)/(factor of leading coefficient). Here, the factors of the constant 6 are 1, 2, 3, 6 and the factors of the leading coefficient 2 are 1, 2. So the candidates are:   ±1, ±2, ±3, ±6, ±1/2, ±3/2. Step 2. Test the candidates. Test x = 1:   P(1) = 2(1)⁴ + 11(1)³ − 8(1)² − 11(1) + 6 = 2 + 11 − 8 − 11 + 6 = 0. Thus, x = 1 is a zero. Step 3. Factor by dividing by (x − 1). Divide P(x) by (x − 1) using synthetic division with the coefficients [2, 11, −8, −11, 6]:   • Bring down 2.   • Multiply by 1: 2   • Add to 11: 13   • Multiply 13 by 1: 13   • Add to −8: 5   • Multiply 5 by 1: 5   • Add to −11: −6   • Multiply −6 by 1: −6   • Add to 6: 0 The quotient is 2x³ + 13x² + 5x − 6. Step 4. Find a zero of the cubic 2x³ + 13x² + 5x − 6. Test x = −1:   Plug in: 2(−1)³ + 13(−1)² + 5(−1) − 6 = −2 + 13 − 5 − 6 = 0. Thus, x = −1 is a zero. Divide the cubic by (x + 1) using synthetic division with −1 and coefficients [2, 13, 5, −6]:   • Bring down 2.   • Multiply by −1: −2   • Add to 13: 11   • Multiply 11 by −1: −11   • Add to 5: −6   • Multiply −6 by −1: 6   • Add to −6: 0 The quotient is now the quadratic 2x² + 11x − 6. Step 5. Solve the quadratic 2x² + 11x − 6 = 0. Use the quadratic formula:   x = [−11 ± √(11² − 4·2·(−6))] / (2·2)     = [−11 ± √(121 + 48)] / 4     = [−11 ± √169] / 4     = [−11 ± 13] / 4. So, we get two solutions:   x = (−11 + 13)/4 = 2/4 = 1/2   x = (−11 − 13)/4 = (−24)/4 = −6. Step 6. List all zeros. The zeros found are:   x = 1, x = −1, x = 1/2, x = −6. All of these zeros are rational. Thus:   Rational zeros: 1, −1, 1/2, −6   Irrational zeros: DNL (Enter "DNL" for "does not exist" since there are no irrational zeros.)