Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the polynomial for each value of \(x\).\(\begin{array}{lll}-x^{2}+3 & \text { (a) } x=-3 & \text { (b) } x=-2\end{array}\) (c) \(x=0\) (d) \(x=1\)

Step-by-Step Solution

Answer

The values of the polynomial at \(x=-3\), \(x=-2\), \(x=0\) and \(x=1\) are -6, -1, 3 and 2 respectively.

Step 1: Evaluate the polynomial at \(x = -3\)

Substitute \(x = -3\) into the function \(-x^{2}+3\): \[ f(-3) = -(-3)^2 + 3 = -9 + 3 = -6 \]

Step 2: Evaluate the polynomial at \(x = -2\)

Substitute \(x = -2\) into the function \(-x^{2}+3\): \[ f(-2) = -(-2)^2 + 3 = -4 + 3 = -1 \]

Step 3: Evaluate the polynomial at \(x = 0\)

Substitute \(x = 0\) into the function \(-x^{2}+3\): \[ f(0) = -0^2 + 3 = 0 + 3 = 3 \]

Step 4: Evaluate the polynomial at \(x = 1\)

Substitute \(x = 1\) into the function \(-x^{2}+3\): \[ f(1) = -(1)^2 + 3 = -1 + 3 = 2 \]

Key Concepts

Polynomial Functions

Polynomial functions represent relationships that involve exponents and coefficients which are usually real numbers. Formally, a polynomial function can be expressed as
\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0 \)
where each \( a_i \) represents a coefficient, \( x \) is the variable, and \( n \) is a non-negative integer that represents the degree of the polynomial. The degree is the highest power of \( x \) that appears in the polynomial.

For example, in the exercise provided, \( -x^2+3 \) is a polynomial function of degree two (since the highest power of \( x \) is two). This means it's a quadratic polynomial, and such polynomials typically have a parabolic shape when graphed. Polynomial functions can take on a variety of forms and provide a mathematical framework for describing numerous types of physical phenomena.

Substitution Method

The substitution method is a means of evaluating an algebraic expression at a specific value of its variable. This involves replacing the variable with the given value and then simplifying the expression.
For evaluating a polynomial, like in our exercise, you follow these steps:

1. Determine the value of the variable for which the polynomial needs to be evaluated.
2. Substitute this value into the polynomial in place of the variable.
3. Simplify the resulting expression using the order of operations - parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS).

in each step of the provided solution, a different value of \( x \) was substituted into the polynomial, and the expression was simplified as demonstrated. This method allows us to find the value of the polynomial function for any real number input for \( x \).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. Unlike equations, they do not have an 'equal' sign and therefore do not express a relationship between two things. Instead, they represent a value that can change depending on the values of its variables.

In the context of the substitution method for polynomials, an algebraic expression is what you get after substituting the numerical values for the variable. For instance, when we substitute \( x = -3 \) into the polynomial function \( -x^2+3 \), we get the algebraic expression \( -(-3)^2+3 \), which simplifies to \( -6 \). Algebraic expressions are essential in helping us understand how to manipulate symbols and numbers to find specific values, which is a foundational skill in algebra.