Find an expression for a cubic function f if f(2) = 36 and f(−4) = f(0) = f(3) = 0. Step 1 A cubic function generally has the fo
rm f(x) = ax3 + bx2 + cx + d. If we know that for some x-value x = p we have f(p) = 0, then it must be true that x − p is a factor of f(x). Since we are told that f(3) = 0, we know that $$ Correct: Your answer is correct. x-3 is a factor.
Step 2 Similarly, since f(−4) = 0, then f(x) has the factor $$ Correct: Your answer is correct. x+4, and since f(0) = 0, then f(x) has the factor (x-0) Correct: Your answer is correct. seenKey x .
Step 3 Since f(x) has the factors x, x − 3, and x + 4, then we must have f(x) = a(x)(x − 3)(x + 4) for some as yet unknown multiplier a. However, since it was given that f(2) = 36, we substitute 2 for x and 36 for f(x) and solve f(2) = 36 = a(2)(2 − 3)(2 + 4) to obtain a = -12 Incorrect: Your answer is incorrect.
Step 2 Similarly, since f(−4) = 0, then f(x) has the factor $$ Correct: Your answer is correct. x+4, and since f(0) = 0, then f(x) has the factor (x-0) Correct: Your answer is correct. seenKey x .
Step 3 Since f(x) has the factors x, x − 3, and x + 4, then we must have f(x) = a(x)(x − 3)(x + 4) for some as yet unknown multiplier a. However, since it was given that f(2) = 36, we substitute 2 for x and 36 for f(x) and solve f(2) = 36 = a(2)(2 − 3)(2 + 4) to obtain a = -12 Incorrect: Your answer is incorrect.
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