1. **Problem Statement:** We have a cubic polynomial $p(x)$. - When divided by $x + 2x - 8$, the remainder is $27x - 45$. - When divided by $(x-2)$, the remainder is $15x$. - $p(x)$ is divisible by $bx + 1$. We need to: (a) Find the quotient when $p(x)$ is divided by $x + 2x - 8$. (b) Determine if $p(x) = 0$ has irrational roots and explain. 2. **Simplify divisor:** Note $x + 2x - 8 = 3x - 8$. 3. **Express $p(x)$ using division algorithm:** When dividing by $3x - 8$, $$p(x) = (3x - 8)Q(x) + (27x - 45)$$ where $Q(x)$ is the quotient polynomial. 4. **Given $p(x)$ is cubic, $Q(x)$ is quadratic:** Let $Q(x) = ax^2 + bx + c$. 5. **Use the remainder when dividing by $(x-2)$:** By the Remainder Theorem, remainder when dividing by $(x-2)$ is $p(2)$. Given remainder is $15x$, which is a polynomial, but remainder must be a constant for linear divisor. This suggests the problem means the remainder polynomial is $15x$ when dividing by some divisor, but since $(x-2)$ is linear, remainder must be constant. Assuming the problem means remainder is $15 imes 2 = 30$ at $x=2$. 6. **Calculate $p(2)$ using the division by $3x - 8$:** $$p(2) = (3(2) - 8)Q(2) + (27(2) - 45) = (6 - 8)Q(2) + (54 - 45) = (-2)Q(2) + 9$$ 7. **Set $p(2) = 30$ (from step 5):** $$(-2)Q(2) + 9 = 30 \\ -2Q(2) = 21 \\ Q(2) = -\frac{21}{2} = -10.5$$ 8. **Express $Q(2)$ in terms of $a,b,c$:** $$Q(2) = a(2)^2 + b(2) + c = 4a + 2b + c = -10.5$$ 9. **Use divisibility by $bx + 1$:** Since $p(x)$ is divisible by $bx + 1$, $p(-\frac{1}{b}) = 0$. 10. **Rewrite $p(x)$ in terms of $Q(x)$:** $$p(x) = (3x - 8)Q(x) + (27x - 45)$$ 11. **Set $p(-\frac{1}{b}) = 0$:** $$0 = (3(-\frac{1}{b}) - 8)Q(-\frac{1}{b}) + (27(-\frac{1}{b}) - 45)$$ 12. **Simplify:** $$0 = \left(-\frac{3}{b} - 8\right)Q\left(-\frac{1}{b}\right) - \frac{27}{b} - 45$$ 13. **Since $p(x)$ is cubic, $Q(x)$ is quadratic:** Let $Q(x) = ax^2 + bx + c$ (reuse $b$ here is confusing, rename coefficients of $Q$ as $A,B,C$): $$Q(x) = Ax^2 + Bx + C$$ 14. **Rewrite $Q(-\frac{1}{b})$:** $$Q\left(-\frac{1}{b}\right) = A\left(-\frac{1}{b}\right)^2 + B\left(-\frac{1}{b}\right) + C = \frac{A}{b^2} - \frac{B}{b} + C$$ 15. **Substitute into equation from step 12:** $$0 = \left(-\frac{3}{b} - 8\right)\left(\frac{A}{b^2} - \frac{B}{b} + C\right) - \frac{27}{b} - 45$$ 16. **We have multiple unknowns $A,B,C,b$.** We need more information or assumptions to solve completely. 17. **Regarding (a) Find the quotient when dividing by $3x - 8$:** We know $Q(x) = Ax^2 + Bx + C$ but cannot find exact coefficients without more data. 18. **Regarding (b) Rationality of roots:** Since $p(x)$ is cubic and divisible by $bx + 1$, one root is $x = -\frac{1}{b}$ (rational if $b$ rational). The other roots come from $Q(x)$ quadratic. If $Q(x)$ has discriminant $$\Delta = B^2 - 4AC$$ negative or non-perfect square, roots are irrational. 19. **Conclusion:** - (a) Quotient is $Q(x) = Ax^2 + Bx + C$ with unknown coefficients. - (b) Without explicit $Q(x)$, cannot confirm if roots are irrational. **Final answer:** (a) Quotient is $Q(x)$ such that $$p(x) = (3x - 8)Q(x) + (27x - 45)$$ with $Q(x)$ quadratic. (b) Cannot confirm irrational roots without more data; roots depend on discriminant of $Q(x)$.