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Chapter 3: Problem 79

Find the value of \(x^{2}-3 x+1\) for each given value of \(x .\) See Section 1.7. $$ 2 $$

Short Answer

Expert verified
The value is -1.

Step by step solution

01

Substitute Given Value

First, we substitute the given value of \(x\) which is 2 into the expression \(x^{2}-3x+1\). This gives us \[(2)^{2} - 3 \cdot 2 + 1.\]
02

Calculate Square and Product

Next, calculate the square of 2 and the product of 3 and 2. We have \[(2)^{2} = 4 \quad \text{and} \quad 3 \cdot 2 = 6.\]
03

Substitute Results

Now substitute the calculated results back into the expression, which becomes \[4 - 6 + 1.\]
04

Perform Addition and Subtraction

Finally, perform the arithmetic operations in the expression: \[4 - 6 + 1 = -1.\]

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Key Concepts

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

Substitution Method
The Substitution Method is a crucial step in evaluating polynomials. It involves replacing variables in an expression with their respective numerical values. In our example, the polynomial expression is \(x^{2} - 3x + 1\), and we want to evaluate it at \(x = 2\).
  • Start by identifying the variable and its given value, which in this case is \(x = 2\).
  • Replace every instance of \(x\) in the polynomial with 2. So, the expression becomes \((2)^2 - 3 \, \times \, 2 + 1\).
Substitution helps simplify the expression, enabling us to directly calculate the value by following straightforward arithmetic steps.
Arithmetic Operations
Arithmetic Operations are the basic mathematical calculations such as addition, subtraction, multiplication, and division. In polynomial evaluation, these operations are performed step by step after substitution.
  • First, tackle the operations inside any parentheses or exponents.
  • In our example, after substituting, we deal with \((2)^2 = 4\).
  • Then, continue with multiplication, \(3 \, \times \, 2 = 6\).
  • Substitute these scores back into the expression, simplifying it to \(4 - 6 + 1\).
  • Finally, use basic addition and subtraction to find the final result: \(4 - 6 + 1 = -1\).
Understanding and following the order of operations is key to evaluating expressions correctly and reaching the right answer.
Exponents
Exponents are a way to express repeated multiplication of a number by itself. They are denoted by a small number written at the top right of a base number, like \(2^2\). In our polynomial \(x^2 - 3x + 1\), \(x^2\) means \(x\) multiplied by itself.
  • An exponent of 2 indicates squaring, so \((2)^2 = 2 \, \times \, 2\).
  • Exponents must be evaluated early in the simplification process, following the order rules in arithmetic, known as "PEMDAS/BODMAS."
  • After evaluating exponents, proceed with other arithmetic operations like multiplication or addition in expressions.
Grasping the concept of exponents ensures that expressions are simplified correctly, leading to an accurate evaluation of polynomial expressions.

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