Suppose f(x) and g(x) are non-zero polynomials with real coefficients, such that f(g(x))=f(x)\cdot g(x). If g(2)=37, what is g(3)?
stop cheating
370
447
Hmmm... hint: You analyse the degree then you're done.
To find the value of g(3), we need to consider the given information:
1. f(x) and g(x) are non-zero polynomials with real coefficients.
2. f(g(x)) = f(x) * g(x).
3. g(2) = 37.
Let's analyze the problem step by step:
Step 1: Start with the equation f(g(x)) = f(x) * g(x).
Step 2: Substituting x = 2 into the equation, we get f(g(2)) = f(2) * g(2).
Step 3: Since g(2) = 37, we can rewrite the equation as f(37) = f(2) * 37.
Step 4: Now, consider the polynomial f(x) as a separate entity.
Step 5: Since f(x) is a non-zero polynomial, it cannot be equal to zero for any value of x.
Step 6: Dividing both sides of the equation f(37) = f(2) * 37 by f(2), we get f(37)/f(2) = 37.
Step 7: Therefore, f(37) = 37 * f(2).
Step 8: From the equation f(37) = f(2) * 37, we notice that f(37) and f(2) must have the same sign (either both positive or both negative) since their product is positive.
Step 9: Now, consider the polynomial g(x) as a separate entity.
Step 10: Since g(x) is also a non-zero polynomial, it cannot be equal to zero for any value of x.
Step 11: Since g(2) = 37, we know that the polynomial g(x) takes the value 37 when x is equal to 2.
Step 12: As g(x) is a polynomial, the function should vary smoothly within the real numbers.
Step 13: Therefore, we can infer that g(x) is not constant and takes a different value when x is different.
Step 14: Since f(37) = 37 * f(2) from Step 7, we can conclude that g(37) = 37 * g(2).
Step 15: Substituting g(2) = 37 into the equation g(37) = 37 * g(2), we get g(37) = 37 * 37.
Step 16: Simplifying the equation, g(37) = 1369.
Step 17: Finally, we have found g(37) = 1369, which implies that g(3) is also equal to 1369 since the polynomial g(x) varies smoothly between values.
Therefore, the value of g(3) is 1369.