Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x^4−3x^2−4). If f(3)=−150, determine the value of a_n.

Verified step by step guidance

1. Substitute \( x = 3 \) into the function \( f(x) = a_n(x^4 - 3x^2 - 4) \) to find \( f(3) \).
2. Set up the equation \( f(3) = a_n(3^4 - 3(3)^2 - 4) = -150 \).
3. Calculate the expression inside the parentheses: \( 3^4 - 3(3)^2 - 4 \).
4. Simplify the expression to find the numerical value.
5. Solve for \( a_n \) by dividing both sides of the equation by the simplified expression.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) is a polynomial of degree 4, which means it can be expressed in the form f(x) = a_n * x^4 + a_(n-1) * x^3 + a_(n-2) * x^2 + ... + a_0. Understanding polynomial functions is crucial for evaluating and manipulating them.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, in the given problem, we need to evaluate f(3) by substituting x with 3 in the polynomial expression. This process is essential for finding the unknown coefficient a_n by setting the evaluated function equal to the given value, -150.

Solving for Unknowns

Solving for unknowns involves using algebraic techniques to find the value of a variable that satisfies an equation. In this context, after evaluating f(3) and setting it equal to -150, we will have an equation in terms of a_n. Rearranging and solving this equation will yield the value of the coefficient a_n, which is a fundamental skill in algebra.