In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x⁴ - x² +1 d. h (3a)

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with evaluating the function h(x) = x⁴ - x² + 1 at the given value of the independent variable, which is 3a. This means substituting 3a for x in the function.
2. Step 2: Substitute 3a into the function h(x). Replace every occurrence of x in the function with 3a. The function becomes h(3a) = (3a)⁴ - (3a)² + 1.
3. Step 3: Simplify the first term (3a)⁴. Use the property of exponents: (ab)⁴ = a⁴b⁴. This gives (3a)⁴ = 3⁴a⁴.
4. Step 4: Simplify the second term (3a)². Similarly, use the property of exponents: (ab)² = a²b². This gives (3a)² = 3²a².
5. Step 5: Combine the simplified terms into the function. Substitute the results from Steps 3 and 4 back into the function: h(3a) = 3⁴a⁴ - 3²a² + 1. You can leave the expression in this simplified form or further evaluate the numerical coefficients (e.g., 3⁴ = 81 and 3² = 9) if needed.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, to evaluate h(3a), we replace x in the function h(x) = x^4 - x² + 1 with 3a. This process allows us to determine the output of the function for that particular input.

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function h(x) = x^4 - x² + 1 is a polynomial of degree 4, which means the highest power of x is 4. Understanding the structure of polynomial functions is essential for evaluating and simplifying them.

Simplification of Expressions

Simplification of expressions involves reducing a mathematical expression to its simplest form. After substituting 3a into h(x), we will need to simplify the resulting polynomial expression by combining like terms and performing any necessary arithmetic operations. This step is crucial for obtaining a clear and concise result.