A biologist is studying a population of beetles whose growth rate follows a cubic function \( f(x) = ax^3 + bx^2 + cx + d \). Observations show \( f(1) = 4 \), \( f(2) = 15 \), \( f(3) = 40 \), and \( f(4) = 85 \). Determine the coefficients \( a \), \( b \), \( c \), and \( d \).

["Unlocking Beetle Population Growth: A Mathematician’s Breakthrough Using Cubic Polynomials", "Understanding population dynamics in biology often relies on mathematical modeling. In a recent study, a biologist investigated the growth pattern of a beetle population over time, identifying that the population size at weeks ( x = 1, 2, 3, 4 ) follows a cubic function:\n[\nf(x) = ax^3 + bx^2 + cx + d\n]\nGiven data points—( f(1) = 4 ), ( f(2) = 15 ), ( f(3) = 40 ), ( f(4) = 85 )—we present a detailed mathematical investigation to determine the coefficients ( a ), ( b ), ( c ), and ( d ), unlocking deeper insights into this ecological system.", "### The Cubic Model Approach", "Cubic polynomials are powerful tools for modeling non-linear growth and decline in biological populations, capturing acceleration and deceleration not described by linear or quadratic functions. Here, fitting ( f(x) ) to the observed data enables precise predictions and understanding of developmental trends.", "Using the known values, we form a system of equations:", "1. ( f(1) = a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 4 )\n2. ( f(2) = 8a + 4b + 2c + d = 15 )\n3. ( f(3) = 27a + 9b + 3c + d = 40 )\n4. ( f(4) = 64a + 16b + 4c + d = 85 )", "### Solving the System of Equations", "We solve this system step-by-step using elimination and substitution.", "Step 1: Eliminate ( d )\nSubtract Equation (1) from the others:", "- Eq2 – Eq1:\n ( (8a + 4b + 2c + d) - (a + b + c + d) = 15 - 4 )\n ( 7a + 3b + c = 11 )  (Equation A)", "- Eq3 – Eq1:\n ( 27a + 9b + 3c + d - (a + b + c + d) = 40 - 4 )\n ( 26a + 8b + 2c = 36 )  (Equation B)", "- Eq4 – Eq1:\n ( 64a + 16b + 4c + d - (a + b + c + d) = 85 - 4 )\n ( 63a + 15b + 3c = 81 )  (Equation C)", "Step 2: Eliminate ( c )\nUse Equation A and B:", "Multiply Equation A by 2:\n( 14a + 6b + 2c = 22 )", "Subtract from Equation B:\n( (26a + 8b + 2c) - (14a + 6b + 2c) = 36 - 22 )\n( 12a + 2b = 14 ) → divide by 2:\n( 6a + b = 7 )  (Equation D)", "Now eliminate ( c ) between Equations B and C:", "Multiply Equation B by 3:\n( 78a + 24b + 6c = 108 )", "Multiply Equation C by 2:\n( 126a + 30b + 6c = 162 )", "Subtract:\n( (126a + 30b + 6c) - (78a + 24b + 6c) = 162 - 108 )\n( 48a + 6b = 54 )\nDivide by 6:\n( 8a + b = 9 )  (Equation E)", "Step 3: Solve for ( a ) and ( b )\nSubtract Equation D from E:\n( (8a + b) - (6a + b) = 9 - 7 )\n( 2a = 2 ) → ( a = 1 )", "Substitute into Equation D:\n( 6(1) + b = 7 ) → ( b = 1 )", "Step 4: Solve for ( c )\nBack-substitute into Equation A:\n( 7(1) + 3(1) + c = 11 ) → ( 7 + 3 + c = 11 ) → ( c = 1 )", "Step 5: Solve for ( d )\nUse Eq1:\n( a + b + c + d = 4 ) → ( 1 + 1 + 1 + d = 4 ) → ( d = 1 )", "### Final Coefficients Summary", "The cubic model for beetle population growth is:\n[\nf(x) = x^3 + x^2 + x + 1\n]\nThus, the coefficients are:\n[\na = 1, \quad b = 1, \quad c = 1, \quad d = 1\n]", "### Interpretation and Biological Insight", "This simple yet elegant cubic function captures precise population trends at discrete time points. The steady linear contribution of ( x ), ( x^2 ), and ( x^3 ) terms suggests accelerating growth patterns consistent with early-stage population expansion under favorable conditions. Such models empower biologists to forecast future population sizes, assess environmental impacts, and design conservation strategies.", "### Conclusion", "By combining empirical data with algebraic modeling, we’ve uncovered the precise cubic function describing beetle population dynamics. This approach exemplifies how math-based analysis drives ecological understanding—turning observations into predictive power. For researchers and educators alike, mastering such techniques illuminates the hidden rhythms of nature, one equation at a time.", "Keywords: cubic polynomial, beetle population, mathematical modeling, biologist data analysis, cubic growth function ( f(x) = ax^3 + bx^2 + cx + d ), solving for coefficients."]