1. **State the problem:** We need to find a cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ such that it passes through the points $(1,24)$, $(3,120)$, $(5,336)$, and $(7,720)$. Then, we will find the value of $p(2)$. 2. **Set up the system of equations:** Using the points, we get: $$\begin{cases} a(1)^3 + b(1)^2 + c(1) + d = 24 \\ a(3)^3 + b(3)^2 + c(3) + d = 120 \\ a(5)^3 + b(5)^2 + c(5) + d = 336 \\ a(7)^3 + b(7)^2 + c(7) + d = 720 \end{cases}$$ 3. **Write the equations explicitly:** $$\begin{cases} a + b + c + d = 24 \\ 27a + 9b + 3c + d = 120 \\ 125a + 25b + 5c + d = 336 \\ 343a + 49b + 7c + d = 720 \end{cases}$$ 4. **Subtract the first equation from the others to eliminate $d$:** $$\begin{cases} (27a - a) + (9b - b) + (3c - c) = 120 - 24 \\ (125a - a) + (25b - b) + (5c - c) = 336 - 24 \\ (343a - a) + (49b - b) + (7c - c) = 720 - 24 \end{cases}$$ which simplifies to: $$\begin{cases} 26a + 8b + 2c = 96 \\ 124a + 24b + 4c = 312 \\ 342a + 48b + 6c = 696 \end{cases}$$ 5. **Divide each equation by the common factor to simplify:** - First equation: divide by 2 $$13a + 4b + c = 48$$ - Second equation: divide by 4 $$31a + 6b + c = 78$$ - Third equation: divide by 6 $$57a + 8b + c = 116$$ 6. **Subtract the first simplified equation from the second and third to eliminate $c$:** - Second minus first: $$ (31a - 13a) + (6b - 4b) + (c - c) = 78 - 48 $$ $$ 18a + 2b = 30 $$ - Third minus first: $$ (57a - 13a) + (8b - 4b) + (c - c) = 116 - 48 $$ $$ 44a + 4b = 68 $$ 7. **Simplify the second equation by dividing by 2:** $$ 9a + b = 15 $$ 8. **Simplify the third equation by dividing by 4:** $$ 11a + b = 17 $$ 9. **Subtract the simplified second from the third to find $a$:** $$ (11a - 9a) + (b - b) = 17 - 15 $$ $$ 2a = 2 $$ $$ a = 1 $$ 10. **Substitute $a=1$ into $9a + b = 15$ to find $b$:** $$ 9(1) + b = 15 $$ $$ 9 + b = 15 $$ $$ b = 6 $$ 11. **Substitute $a=1$, $b=6$ into $13a + 4b + c = 48$ to find $c$:** $$ 13(1) + 4(6) + c = 48 $$ $$ 13 + 24 + c = 48 $$ $$ 37 + c = 48 $$ $$ c = 11 $$ 12. **Substitute $a=1$, $b=6$, $c=11$ into $a + b + c + d = 24$ to find $d$:** $$ 1 + 6 + 11 + d = 24 $$ $$ 18 + d = 24 $$ $$ d = 6 $$ 13. **Write the cubic polynomial:** $$ p(x) = x^3 + 6x^2 + 11x + 6 $$ 14. **Find $p(2)$:** $$ p(2) = 2^3 + 6(2^2) + 11(2) + 6 = 8 + 6(4) + 22 + 6 = 8 + 24 + 22 + 6 = 60 $$ **Final answer:** The cubic polynomial is $p(x) = x^3 + 6x^2 + 11x + 6$ and $p(2) = 60$.