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Chapter 8: Problem 98

Find the remainder when \(x^{8}-2 x+1\) is divided by \(x-2\).

Short Answer

Expert verified
The remainder is 253.

Step by step solution

01

Understand the Problem

The task is to find the remainder when the polynomial \(x^{8} - 2x + 1\) is divided by \(x - 2\). This can be solved using the Remainder Theorem.
02

Apply the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \(f(x)\) by \(x - c\) is \(f(c)\). In this case, \(f(x) = x^{8} - 2x + 1\) and \(c = 2\).
03

Substitute \(x = 2\) into the Polynomial

Substitute \(x = 2\) into \(f(x) = x^{8} - 2x + 1\): \( f(2) = 2^{8} - 2(2) + 1 \).
04

Calculate the Expression

Calculate the value of \(2^{8} - 2(2) + 1\): \(2^{8} = 256\), \(2(2) = 4\), so \(256 - 4 + 1 = 253\).
05

State the Remainder

The remainder when \(x^{8} - 2x + 1\) is divided by \(x - 2\) is 253.

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Key Concepts

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

Polynomial Division
Polynomial division is a method used to divide one polynomial by another. It functions similarly to long division with numbers and helps break down complex expressions.

When dividing a polynomial, the divisor (the polynomial you are dividing by) and the dividend (the polynomial being divided) are involved. The quotient (result) and the remainder are calculated.

For example, dividing \(x^2 + 3x + 2\) by \(x + 1\) involves:
  • Dividing the leading term of the dividend by the leading term of the divisor.
  • Multiplying the whole divisor by the quotient obtained in each step.
  • Subtracting this product from the dividend and bringing down the next terms.
This process continues until all terms are divided, resulting in a quotient and possibly a remainder.
Evaluating Polynomials
Evaluating a polynomial means finding its value at a given point. This involves substituting the variable in the polynomial with a specific number.

For instance, to evaluate the polynomial \(f(x) = x^2 - 3x + 2\) at \(x = 1\):
  • Substitute \(1\) for \(x\): \(f(1) = 1^2 - 3(1) + 2\).
  • Simplify: \(1 - 3 + 2 = 0\).
This results in \(f(1) = 0\).

Evaluating polynomials is crucial in applying the Remainder Theorem. It helps determine the remainder when a polynomial is divided by another polynomial of the form \(x - c\) by simply substituting \(x = c\) into the polynomial.
Remainder Calculation
The Remainder Theorem provides a quick way to find the remainder of a polynomial divided by a linear divisor \(x - c\).

According to the theorem, the remainder of the division of \(f(x)\) by \(x - c\) is \(f(c)\).

For example, to find the remainder when \(x^8 - 2x + 1\) is divided by \(x - 2\):
  • Identify the polynomial: \(f(x) = x^8 - 2x + 1\).
  • Set \(c = 2\).
  • Evaluate the polynomial at \(x = 2\): \(f(2) = 2^8 - 2(2) + 1\).

Calculate the values:
  • \(2^8 = 256\)
  • \(2(2) = 4\)
The remainder is \(256 - 4 + 1 = 253\).

Thus, the remainder when \(x^8 - 2x + 1\) is divided by \(x - 2\) is 253.

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Most popular questions from this chapter

Use a graphing calculator to graph the equation corresponding to each inequality. From the display of the graphing calculator, locate one ordered pair in the solution set to the system and check that it satisfies all inequalities of the system. Answers may vary. $$\begin{aligned}&y2^{x}\\\&y<5 x-6\end{aligned}$$

Sketch the graph of the solution set to each nonlinear system of inequalities. $$\begin{aligned}&x^{2}+y^{2}<36\\\&|y|<3\end{aligned}$$

Find a system of inequalities to describe the given region. Points in the first quadrant less than nine units from the origin.

Find the equation of the axis of symmetry for the graph of \(g(x)=-3 x^{2}-5 x+9\).

What is the equation of the horizontal asymptote to the graph of \(f(x)=3 e^{x-4}+5 ?\)
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