Answer :
Final answer:
To find the values of d and f, use the Remainder Theorem and set up equations based on the given remainders.
Explanation:
To find the values of d and f, we can use the Remainder Theorem. In this case, h(x) is divided by (x - 1) and (x - 3) with remainders of -10 and 36 respectively.
Using the Remainder Theorem, we can set up two equations:
h(1) = d(1)^3 + f(1)^2 - 7(1) - 6 = -10
h(3) = d(3)^3 + f(3)^2 - 7(3) - 6 = 36
Solving these equations will give us the values of d and f.
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Final answer:
The problem can be solved using the Remainder theorem which gives us two linear equations in terms of d and f. Solving this system of equations yields the values of constants d and f.
Explanation:
To solve this problem, we use the Remainder Theorem, which says that the remainder of a polynomial p(x) when divided by (x-c) is p(c). So we can express our given parameters as h(1) = -10 and h(3) = 36.
Substitute these values into our polynomial equation and it gives us two equations, like this:
1. For h(1) = -10: d(1)^3 + f(1)^2 - 7(1) - 6 = -10, which simplifies to d + f - 7 - 6 = -10, or d + f = -10 + 7 + 6, so d + f = 3.
2. Similarly, for h(3) = 36: d(3)^3 + f(3)^2 - 7(3) - 6 = 36, simplifying to 27d + 9f - 21 - 6 = 36, or 27d + 9f = 36 + 21 + 6, so 27d + 9f = 63.
Now you can solve these two equations to find the values of constants d and f. This is a basic system of two linear equations that can be solved by substitution or elimination method.
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