{Use of Tech} Polynomial calculations
Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)
ƒ(ƒ(x)) = x⁴ - 12x² + 30

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Determine the degree of the polynomial \( f(x) \). Since \( f(f(x)) = x^4 - 12x^2 + 30 \), and the degree of \( f(f(x)) \) is the square of the degree of \( f(x) \), the degree of \( f(x) \) must be 2.

Assume \( f(x) = ax^2 + bx + c \).

Substitute \( f(x) \) into \( f(f(x)) \) to get \( f(f(x)) = a(ax^2 + bx + c)^2 + b(ax^2 + bx + c) + c \).

Expand \( (ax^2 + bx + c)^2 \) and substitute back into the expression for \( f(f(x)) \).

Equate the coefficients of \( f(f(x)) \) to those of \( x^4 - 12x^2 + 30 \) to solve for \( a, b, \) and \( c \).

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주요 개념

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Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is ƒ(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer representing the degree of the polynomial. Understanding polynomial functions is crucial for manipulating and solving equations involving them.

Composition of Functions

The composition of functions involves combining two functions where the output of one function becomes the input of another. In this context, ƒ(ƒ(x)) means applying the polynomial function ƒ to itself. This concept is essential for solving the equation given, as it requires understanding how to manipulate and evaluate the polynomial when substituted into itself.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It determines the polynomial's behavior and the number of roots it can have. In this problem, identifying the degree of the polynomial ƒ is critical, as it guides the selection of a suitable polynomial form to substitute and solve for the coefficients that satisfy the equation ƒ(ƒ(x)) = x⁴ - 12x² + 30.

{Use of Tech} Polynomial calculationsFind a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)ƒ(ƒ(x)) = x⁴ - 12x² + 30