Problem 28 - Precalculus : Building Concepts and Connections

Question

Use synthetic division to find the function values. \(f(x)=-x^{4}+3 x^{2}-2 x ;\) find \(f\left(\frac{3}{2}\right)\)

Step-by-Step Solution

Answer

The function value of \( f(\frac{3}{2}) \) is -7.3125

Step 1: Substitute the x in the function with \( \frac{3}{2} \)

Replace each instance of the variable 'x' in the function \( f(x) = -x^{4} + 3x^{2} - 2x \) with the value \( \frac{3}{2} \). The function then becomes \( f\left( \frac{3}{2} \right) = -\left( \frac{3}{2} \right)^{4} + 3\left( \frac{3}{2} \right)^2 - 2\left( \frac{3}{2} \right)

Step 2: Simplify each term

Simplify each term in the equation separately before trying to bring them together. This would result in \( f\left(\frac{3}{2}\right) = -\left(\frac{81}{16}\right) + \left(\frac{27}{4}\right) - \left(\frac{3}{1}\right)

Step 3: Subtract all terms

Now you can compute the final calculation. That is, subtract all the terms from each other. This simplifies to \( f(\frac{3}{2}) = -\frac{81}{16} + \frac{27}{4} - \frac{3}{1} = -\frac{117}{16} = -7.3125 \)

Key Concepts

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions used in mathematics. They are composed of terms consisting of a coefficient (a number) multiplied by a variable (like x) raised to a non-negative integer exponent. For example, in the function f(x) = -x⁴ + 3x² - 2x, there are three terms, and the largest exponent is 4, which makes it a quartic polynomial.

One of the main features of polynomial functions is their smooth, continuous shape, which makes them very predictable and easy to work with. They can represent a wide range of phenomena including trajectories in physics, trends in economics, and models in engineering. To understand these functions, it is crucial to grasp how to perform operations with them, such as addition, subtraction, and particularly, how to evaluate them, which is the process of finding the value of a polynomial for a specific value of x.

Function Evaluation

Evaluating a function means finding out what its output is for a certain input. In the case of our polynomial function f(x), we want to determine the result when x is \(\frac{3}{2}\). To evaluate f(x) at \(\frac{3}{2}\), each occurrence of x in the function gets replaced with \(\frac{3}{2}\).

However, replacing x with a fraction requires careful simplification; each term with the substituted value becomes a fraction itself, which then must be evaluated through arithmetic operations. In our example, once we substitute x with \(\frac{3}{2}\), we must raise it to the appropriate powers and then multiply by the coefficients, such as in -\(\left(\frac{3}{2}\right)^4\). The next step involves simplifying these terms before combining them to find the overall function value. This process of function evaluation is commonly performed in algebra and calculus and is essential for analyzing function behavior.

Simplifying Expressions

Simplifying expressions is a process often used alongside function evaluation to make equations easier to understand and solve. After substituting in the value of x, as we did with \(\frac{3}{2}\) in our polynomial function, we simplified each term before combining them. This simplification is important because it allows us to deal with simpler numbers or expressions before tackling the entire equation.

For the given function f\(\left(\frac{3}{2}\right)\), each term becomes a fraction after substitution. We simplify these fraction terms by performing exponentiation first, then multiplication with the coefficients, and finally, arithmetic operations between the terms. The simplification process involves reducing fractions, combining like terms, and following the order of operations, known as PEMDAS (parentheses, exponents, multiplication and division, and addition and subtraction). Simplifying correctly is vital as it leads to the correct final value of the function, which in this scenario resulted in -\(\frac{117}{16}\) or -7.3125 when expressed as a decimal.