Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x⁴−3x²−4). If f(3)=−150, determine the value of a_n.

Verified step by step guidance

1. Start with the given function: $f\left(x\right)=a{n}_{}(x^4-3x^2-4)$.
2. Substitute $x=3$ into the function to use the given value $f\left(3\right)=\; -150$. This gives: $-150\; =a{n}_{}(3^4-\; 33^2-\; 4)$.
3. Calculate the powers of 3 inside the parentheses: $3^4=\; 81$ and $3^2=\; 9$.
4. Simplify the expression inside the parentheses: $81\; -\; 3\; imes\; 9\; -\; 4$. Perform the multiplication and subtraction step-by-step.
5. After simplifying the parentheses, solve for $a{n}_{}$ by dividing both sides of the equation by the simplified value inside the parentheses.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into the function's expression to find the corresponding output. In this problem, you substitute x = 3 into f(x) to find f(3), which helps in solving for the unknown coefficient a_n.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding the structure of the polynomial f(x) = a_n(x^4 − 3x^2 − 4) is essential to correctly substitute values and manipulate the equation.

Solving for an Unknown Coefficient

When a function includes an unknown coefficient, you can find its value by using given function values. Here, knowing f(3) = -150 allows you to set up an equation and solve for a_n by isolating it after substituting x = 3 into the polynomial.