Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the polynomial for each value of \(x\).\(-2 x^{2}+3 x+4 \quad\) (a) \(x=-2 \quad\) (b) \(x=-1\) (c) \(x=0\) (d) \(x=1\)

Step-by-Step Solution

Answer

The results of the polynomial for the given values of \(x\) are: (a) -10, (b) -1, (c) 4, and (d) 5.

Step 1: Substitution for \(x = -2\)

Substitute \(x = -2\) into the polynomial \( -2x^2 +3x + 4 \) . This gives: \(-2(-2)^2 +3(-2) + 4 = -8 - 6 + 4 = -10 \)

Step 2: Substitution for \(x = -1\)

Substitute \(x = -1\) into the polynomial \( -2x^2 +3x + 4 \). This gives: \(-2(-1)^2 + 3(-1) + 4 = -2 - 3 + 4 = -1\)

Step 3: Substitution for \(x = 0\)

Substitute \(x = 0\) into the polynomial \( -2x^2 +3x + 4 \). This gives: \(-2(0)^2 + 3(0) + 4 = 0 + 0 + 4 = 4 \)

Step 4: Substitution for \(x = 1\)

Substitute \(x = 1\) into the polynomial \( -2x^2 +3x + 4 \). This gives: \(-2(1)^2 + 3(1) + 4 = -2 + 3 + 4 = 5 \)

Key Concepts

Substitution Method

The substitution method is a mathematical technique used to evaluate polynomials by replacing variables with specific values. This method makes it easy to see how a polynomial behaves for different inputs. Here's how it works:

\(x\) is replaced by the specific value you want to evaluate in the polynomial expression. For example, if you have a polynomial like \(-2x^2 + 3x + 4\), and you want to evaluate it for \(x = -2\), you substitute \(-2\) for \(x\) everywhere in the equation.

• Substitute the given value into the polynomial.
• Perform the arithmetic operations following the standard order: first exponents, then multiplication/division, and finally addition/subtraction.
• Simplify the expression to find the result.

This method can be repeated for any value, allowing you to understand the effect of different \(x\) values on the polynomial.

Quadratic Polynomial

A quadratic polynomial is a type of polynomial with a degree of 2. Its general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In the quadratic polynomial \(-2x^2 + 3x + 4\), we have:

• \(-2x^2\): The quadratic term, where \(-2\) is the coefficient.
• \(3x\): The linear term, with \(3\) as the coefficient.
• \(+4\): The constant term.

Quadratic polynomials are essential in many scientific fields, including physics and engineering, as they can describe trajectories and model various phenomena. Solving a quadratic polynomial typically involves evaluating it at various points or finding its roots, where the polynomial equals zero. Understanding each component helps in analyzing the polynomial's behavior.

Function Evaluation

Function evaluation is the process of computing the output of a function for given input values. When you evaluate a polynomial function, you use known input values to find the corresponding output. For quadratic polynomials, this process aids in analyzing graphs, understanding vertex locations, and predicting values.

Here's how to evaluate a polynomial function like \(-2x^2 + 3x + 4\):

• Identify the function's equation.
• Select a specific value of \(x\) to evaluate.
• Substitute this value into the equation.
• Perform the calculation to find the result.

Evaluating a function answers the basic question: What is the output for a specific input? In more complex scenarios, this process can help deduce trends, solve equations, or analyze systems.