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Chapter 3: Problem 65

The remainder theorem indicates that when a polynomial \(f(x)\) is divided by \(x-k\) the remainder is equal to \(f(k) .\) For $$f(x)=x^{3}-2 x^{2}-x+2$$ use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of \(f(x)\) $$f(2)$$

Short Answer

Expert verified
The remainder is 0. The point on the graph is (2, 0).

Step by step solution

01

- Identify Polynomial and Value

Identify the given polynomial and the value at which to evaluate it. The polynomial is: \[ f(x) = x^3 - 2x^2 - x + 2 \] and you are asked to find \( f(2) \).
02

- Substitute the Value into the Polynomial

Substitute \( x = 2 \) into the polynomial: \[ f(2) = (2)^3 - 2(2)^2 - (2) + 2 \]
03

- Evaluate the Expression

Now, calculate each term step by step: \[ f(2) = 8 - 2 \times 4 - 2 + 2 \] This simplifies to: \[ f(2) = 8 - 8 - 2 + 2 \]
04

- Simplify the Result

Combine the terms to find the remainder: \[ f(2) = 8 - 8 - 2 + 2 = 0 \]
05

- Determine the Coordinates

The remainder given by the Remainder Theorem is the value of the polynomial at the given point. Therefore, the coordinates on the graph of \( f(x) \) corresponding to this remainder are: \( (2, f(2)) = (2, 0) \).

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Key Concepts

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

Polynomial Division
Polynomial division helps us break down complex polynomials into simpler parts. It works similar to long division with numbers. The main goal is to determine what you get when you divide one polynomial by another. In this exercise, we use the Remainder Theorem:

The theorem states that when a polynomial \(f(x)\) is divided by \(x-k\), the remainder is \(f(k)\). For example, if you divide \(f(x) = x^3 - 2x^2 - x + 2\) by \(x-2\), you can evaluate \(f(2)\) to find the remainder.

This makes the process simpler and faster since you don鈥檛 have to perform full polynomial division. Instead, you just substitute and calculate. This method can be applied to any polynomial and value of \(k\).
Evaluating Polynomials
Evaluating polynomials involves substituting a specific value into the polynomial and calculating the result. This helps in finding the output of the polynomial for any given input.

In our example, we evaluated \(f(x) = x^3 - 2x^2 - x + 2\) at \(x = 2\). Here鈥檚 how it works step by step:
  • Identify the polynomial and value: \(f(x) = x^3 - 2x^2 - x + 2\) and \(f(2)\).
  • Substitute \(x = 2\): \(f(2) = (2)^3 - 2(2)^2 - 2 + 2\).
  • Calculate the individual terms: \(f(2) = 8 - 2 \times 4 - 2 + 2\).
  • Combine terms: \(f(2) = 8 - 8 - 2 + 2 = 0\).
The result is \(f(2) = 0\), showing that when \(x = 2\), the value of the polynomial function is zero.
Graph Coordinates
Finding graph coordinates involves determining points on the Cartesian plane where the graph of the polynomial exists. When you know the value of a polynomial function at a specific input, you can plot that point on a graph.

For our polynomial \(f(x) = x^3 - 2x^2 - x + 2\) and input \(x = 2\), we found that \(f(2) = 0\). This gives us the point \((2, 0)\) on the graph. Here鈥檚 how to interpret it:
  • The x-coordinate is the input value, in this case, \(2\).
  • The y-coordinate is the result of the polynomial function, which is \(f(2) = 0\).
  • Therefore, the point \((2, 0)\) is where the polynomial crosses the x-axis.
This method helps in understanding the behavior of the polynomial at different points and contributes to sketching an accurate graph.

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Most popular questions from this chapter

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The remainder theorem indicates that when a polynomial \(f(x)\) is divided by \(x-k\) the remainder is equal to \(f(k) .\) For $$f(x)=x^{3}-2 x^{2}-x+2$$ use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of \(f(x)\) $$f(0)$$
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