1. **State the problem:** Factor the cubic expression $$2x^3 - 13x - 24$$. 2. **Recall the factoring approach:** For cubic polynomials, try to find rational roots using the Rational Root Theorem, then factor by polynomial division or synthetic division. 3. **Possible rational roots:** Factors of constant term 24 over factors of leading coefficient 2: $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm24, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{4}{2}$$ (simplified). 4. **Test roots:** Substitute values into $$2x^3 - 13x - 24$$. - For $$x=3$$: $$2(3)^3 - 13(3) - 24 = 2(27) - 39 - 24 = 54 - 63 = -9 \neq 0$$. - For $$x=-3$$: $$2(-3)^3 - 13(-3) - 24 = 2(-27) + 39 - 24 = -54 + 15 = -39 \neq 0$$. - For $$x=4$$: $$2(4)^3 - 13(4) - 24 = 2(64) - 52 - 24 = 128 - 76 = 52 \neq 0$$. - For $$x=-4$$: $$2(-4)^3 - 13(-4) - 24 = 2(-64) + 52 - 24 = -128 + 28 = -100 \neq 0$$. - For $$x=\frac{3}{2}$$: $$2\left(\frac{3}{2}\right)^3 - 13\left(\frac{3}{2}\right) - 24 = 2\left(\frac{27}{8}\right) - \frac{39}{2} - 24 = \frac{54}{8} - \frac{39}{2} - 24 = \frac{27}{4} - \frac{39}{2} - 24$$. Convert to common denominator 4: $$\frac{27}{4} - \frac{78}{4} - \frac{96}{4} = \frac{27 - 78 - 96}{4} = \frac{-147}{4} \neq 0$$. - For $$x=-\frac{3}{2}$$: $$2\left(-\frac{3}{2}\right)^3 - 13\left(-\frac{3}{2}\right) - 24 = 2\left(-\frac{27}{8}\right) + \frac{39}{2} - 24 = -\frac{54}{8} + \frac{39}{2} - 24 = -\frac{27}{4} + \frac{78}{4} - \frac{96}{4} = \frac{-27 + 78 - 96}{4} = \frac{-45}{4} \neq 0$$. - For $$x=8$$: $$2(8)^3 - 13(8) - 24 = 2(512) - 104 - 24 = 1024 - 128 = 896 \neq 0$$. - For $$x=-8$$: $$2(-8)^3 - 13(-8) - 24 = 2(-512) + 104 - 24 = -1024 + 80 = -944 \neq 0$$. 5. **Try factoring by grouping or guess factors:** The options suggest factors of the form $$(2x \pm 3)(x \pm 8)$$ or $$(2x \pm 3)(x \pm 8)$$. 6. **Expand each option to check:** - Option (2x - 8)(x - 3): $$2x \cdot x = 2x^2$$ (not cubic), discard. - Option (x + 3)(2x - 8): $$x \cdot 2x = 2x^2$$ (not cubic), discard. - Option (2x + 3)(x - 8): $$2x \cdot x = 2x^2$$ (not cubic), discard. - Option (2x + 3)(x + 8): $$2x \cdot x = 2x^2$$ (not cubic), discard. 7. **Since none of the given options produce a cubic term, the expression is not factored correctly by any of these.** 8. **Try synthetic division with root 3:** Coefficients: 2 (x^3), 0 (x^2), -13 (x), -24 (constant) Synthetic division by 3: - Bring down 2 - Multiply 2*3=6, add to 0: 6 - Multiply 6*3=18, add to -13: 5 - Multiply 5*3=15, add to -24: -9 (remainder) Not zero remainder, so 3 is not root. 9. **Try synthetic division with root -3:** - Bring down 2 - Multiply 2*(-3) = -6, add to 0: -6 - Multiply -6*(-3) = 18, add to -13: 5 - Multiply 5*(-3) = -15, add to -24: -39 (remainder) Not zero remainder. 10. **Try root 4:** - Bring down 2 - Multiply 2*4=8, add to 0: 8 - Multiply 8*4=32, add to -13: 19 - Multiply 19*4=76, add to -24: 52 (remainder) No. 11. **Try root -4:** - Bring down 2 - Multiply 2*(-4) = -8, add to 0: -8 - Multiply -8*(-4) = 32, add to -13: 19 - Multiply 19*(-4) = -76, add to -24: -100 No. 12. **Try root 1:** - Bring down 2 - Multiply 2*1=2, add to 0: 2 - Multiply 2*1=2, add to -13: -11 - Multiply -11*1=-11, add to -24: -35 No. 13. **Try root -1:** - Bring down 2 - Multiply 2*(-1) = -2, add to 0: -2 - Multiply -2*(-1) = 2, add to -13: -11 - Multiply -11*(-1) = 11, add to -24: -13 No. 14. **Try root 6:** - Bring down 2 - Multiply 2*6=12, add to 0: 12 - Multiply 12*6=72, add to -13: 59 - Multiply 59*6=354, add to -24: 330 No. 15. **Try root -6:** - Bring down 2 - Multiply 2*(-6) = -12, add to 0: -12 - Multiply -12*(-6) = 72, add to -13: 59 - Multiply 59*(-6) = -354, add to -24: -378 No. 16. **Try root 8:** - Bring down 2 - Multiply 2*8=16, add to 0: 16 - Multiply 16*8=128, add to -13: 115 - Multiply 115*8=920, add to -24: 896 No. 17. **Try root -8:** - Bring down 2 - Multiply 2*(-8) = -16, add to 0: -16 - Multiply -16*(-8) = 128, add to -13: 115 - Multiply 115*(-8) = -920, add to -24: -944 No. 18. **Try root 12:** - Bring down 2 - Multiply 2*12=24, add to 0: 24 - Multiply 24*12=288, add to -13: 275 - Multiply 275*12=3300, add to -24: 3276 No. 19. **Try root -12:** - Bring down 2 - Multiply 2*(-12) = -24, add to 0: -24 - Multiply -24*(-12) = 288, add to -13: 275 - Multiply 275*(-12) = -3300, add to -24: -3324 No. 20. **Try root 24:** - Bring down 2 - Multiply 2*24=48, add to 0: 48 - Multiply 48*24=1152, add to -13: 1139 - Multiply 1139*24=27336, add to -24: 27312 No. 21. **Try root -24:** - Bring down 2 - Multiply 2*(-24) = -48, add to 0: -48 - Multiply -48*(-24) = 1152, add to -13: 1139 - Multiply 1139*(-24) = -27336, add to -24: -27360 No. 22. **Try root 1/2:** - Bring down 2 - Multiply 2*0.5=1, add to 0: 1 - Multiply 1*0.5=0.5, add to -13: -12.5 - Multiply -12.5*0.5=-6.25, add to -24: -30.25 No. 23. **Try root -1/2:** - Bring down 2 - Multiply 2*(-0.5) = -1, add to 0: -1 - Multiply -1*(-0.5) = 0.5, add to -13: -12.5 - Multiply -12.5*(-0.5) = 6.25, add to -24: -17.75 No. 24. **Since no rational roots found, the polynomial is prime over rationals and cannot be factored into linear factors with integer coefficients.** 25. **Conclusion:** None of the given options correctly factor the expression $$2x^3 - 13x - 24$$. **Final answer:** None of the provided factorizations are correct.