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Chapter 5: Problem 23

For each given polynomial, find the indicated value of the polynomial. $$P(x)=x^{4}-1, \quad P(3)$$

Short Answer

Expert verified
P(3) = 80

Step by step solution

01

- Understand the Problem

The polynomial given is \(P(x) = x^4 - 1\) and you need to find \(P(3)\). This means substituting \(x\) with 3 in the polynomial equation.
02

- Substitute the Value

Substitute \(x = 3\) into the polynomial \(P(x)\). So, \(P(3) = 3^4 - 1\).
03

- Calculate the Power

Calculate \(3^4\). You get \(3^4 = 81\).
04

- Subtract One

Subtract 1 from 81. So, \(81 - 1 = 80\).
05

- Write the Final Answer

Thus, \(P(3) = 80\).

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Key Concepts

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

polynomial functions
Polynomial functions are equations that involve variables raised to non-negative integer powers. These functions consist of terms like \(x^2\), \(x^3\), and so on, combined by addition, subtraction, or multiplication. Each term has a coefficient (a constant number) and a variable part. For example, in the polynomial \(2x^3 + 3x^2 - 5x + 6\),
  • 2 is the coefficient of the term \(2x^3\),
  • 3 is the coefficient of the term \(3x^2\),
  • -5 is the coefficient of the term \(-5x\),
  • and 6 is a constant.
Polynomial functions are foundational in algebra and calculus, where they model a variety of complex real-world phenomena.
substitution method
The substitution method involves replacing a variable with a given number, allowing you to evaluate the polynomial at that specific point. In our example, we need to evaluate \(P(x) = x^4 - 1\) at \(x = 3\). This method is simple and follows these steps:
  • Identify the variable in the polynomial (\(x\) in this case).
  • Replace this variable with the given number.
  • Perform the necessary arithmetic operations to find the value.
For instance, substituting \(x = 3\) into the polynomial \(x^4 - 1\), we get \(3^4 - 1\). The substitution method is powerful and is commonly used in algebra to solve equations and evaluate expressions.
exponentiation
Exponentiation is the mathematical operation involving raising a number to the power of another number. In our example, we encountered \(3^4\), where 3 is the base and 4 is the exponent. To solve \(3^4\), you multiply the base by itself as many times as indicated by the exponent:
  • \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)
Understanding exponentiation is crucial because it helps simplify complex calculations and is used extensively in various areas of math and science.
arithmetic operations
Arithmetic operations are the basic mathematical operations which include addition, subtraction, multiplication, and division. In our example, we encountered both exponentiation and subtraction. After calculating the exponentiation \(3^4 = 81\), we performed subtraction:
  • \(81 - 1 = 80\)
These operations are fundamental and allow us to simplify and solve polynomial expressions effectively. Mastering these operations is essential for handling more advanced problems in mathematics.

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