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Chapter 1: Problem 97

Evaluate \(x^{2}+3 x-4\) when \(x=-3 .[\text { A. } 2]\)

Short Answer

Expert verified
The value of the expression \(x^{2}+3 x-4\) when \(x=-3\) is -4.

Step by step solution

01

Substitute the given value of x

Replace \(x\) in the equation \(x^{2}+3 x-4\) with -3. So, the equation becomes \((-3)^{2}+3(-3)-4\)
02

Calculate the power

The equation \((-3)^{2}+3(-3)-4\) becomes \(9+3(-3)-4\) after calculating \((-3)^{2}\)
03

Calculate the multiplication

Multiply 3 with -3. The expression now becomes \(9+(-9)-4\)
04

Perform Addition and Subtraction

Performing the addition and subtraction operations, we get \(9+(-9)-4 = -4\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
Substitution is a fundamental technique in algebra used to evaluate expressions. It involves replacing a variable with a specific value. This method is particularly useful when working with polynomial functions. In the given exercise, the substitution method helps us determine the value of the polynomial function when \(x = -3\).

The main steps of this method include:
  • Rewriting the original polynomial function by substituting the given value for \(x\).
  • Ensuring accuracy by replacing all instances of \(x\) in the polynomial expression.
  • Following up with additional arithmetic operations to simplify the expression to a single numerical result.
By following these steps, you simplify the function, thus making it easier to evaluate and solve.
Polynomial Functions
Polynomial functions form the backbone of many algebraic concepts. They consist of terms that express powers of variables summed together typically in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Each polynomial term has a degree, represented by the power of \(x\).

In our specific polynomial \(x^2 + 3x - 4\), we see several essential components:
  • Degrees: The highest power here is 2, indicating that it is a quadratic equation.
  • Coefficients: They are the numerical factors beside each term, in this case, 1 for \(x^2\), 3 for \(x\), and -4 for the constant.
  • Constant Terms: These are numbers without a variable, like the -4 in our expression.
Understanding these elements helps in evaluating and graphing polynomial functions, as well as predicting their behavior.
Order of Operations
The order of operations is a set of rules that indicates the order in which to perform mathematical operations, ensuring consistency in calculations. This order is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

For the given polynomial expression \((-3)^2 + 3(-3) - 4\), the order of operations is applied as follows:
  • Exponents: Calculate \((-3)^2\) first, resulting in 9.
  • Multiplication: Perform the multiplication \(3 \cdot (-3)\), which yields -9.
  • Addition and Subtraction: Finally, conduct operations from left to right: 9 + (-9) - 4, resulting in -4.
Following the order of operations precisely avoids errors and ensures that the evaluation of expressions is accurate.

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