Problem 57 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. What is the remainder when \(P(x)=2 x^{4}-3 x^{3}-x+7\) is divided by \(x+2 ?\)

Step-by-Step Solution

Answer

The remainder is 65.

Step 1: Understand the Problem

The task is to find the remainder when the polynomial \(P(x) = 2x^{4} - 3x^{3} - x + 7\) is divided by \(x + 2\). To do this, the Remainder Theorem will be used, which states that the remainder of the division of a polynomial \(P(x)\) by \(x - c\) is \(P(c)\).

Step 2: Rewrite the Divisor

The polynomial \(P(x)\) is being divided by \(x + 2\). Using the Remainder Theorem, rewrite \(x + 2\) in the form \(x - (-2)\). Hence, \(c = -2\).

Step 3: Substitute \( c = -2 \) into \(P(x)\)

Substitute \(-2\) for \(x\) in the polynomial \(P(x) = 2x^4 - 3x^3 - x + 7\): \[ P(-2) = 2(-2)^4 - 3(-2)^3 - (-2) + 7 \]

Step 4: Simplify the Expression

Calculate each term of the polynomial with \(x = -2\): \[ 2(-2)^4 = 2(16) = 32 \] \[ -3(-2)^3 = -3(-8) = 24 \] \[ -(-2) = 2 \] \[ 7 = 7 \] Then combine these results: \[ P(-2) = 32 + 24 + 2 + 7 \]

Step 5: Add the Results Together

Sum the values obtained from the previous step: \[ 32 + 24 + 2 + 7 = 65 \] Therefore, the remainder when \(P(x)\) is divided by \(x + 2\) is \(65\).

Key Concepts

Polynomial Division

Polynomial Division is a technique used to divide one polynomial by another. Imagine dividing numbers normally; polynomial division is similar but a bit more complicated.

It can be broken down into manageable steps:

• Organize the polynomials in standard form (highest to lowest degree).
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the whole divisor by this result and subtract from the original polynomial.
• Repeat the process with the new polynomial created after subtraction until the remainder's degree is less than the divisor's degree.

While this traditional method works well, the Remainder Theorem provides a shortcut for finding just the remainder quickly.

Algebra

Algebra is the part of mathematics that deals with variables to represent unknown values. Basic algebra involves balancing equations and manipulating expressions to find the value of these variables.

In this exercise, we're leveraging algebra to simplify the polynomial and find the remainder. By substituting specific values into the polynomial equation, we can determine the outcome without performing lengthy division. This technique is particularly handy for polynomials with higher degrees.

Using algebra to manipulate and simplify complex expressions is fundamental in solving polynomial division problems efficiently. Mastering these algebraic manipulations allows you to approach problems from various angles and find solutions more effectively.

Remainder Calculation

Remainder Calculation in polynomials can be simplified by using the Remainder Theorem. This theorem states that if you divide a polynomial, say, \(P(x)\) by \(x - c\), the remainder of this division is equal to \(P(c)\).

In our example, the problem asked for the remainder when the polynomial \(P(x) = 2x^4 - 3x^3 - x + 7\) is divided by \(x + 2\). Using the Remainder Theorem:

• First, rewrite the divisor in the form of \(x - c\), making \(c = -2\).
• Substitute \(-2\) into the polynomial, replacing every \(x\) with \(-2\).
• Simplify the resulting expression.
• The final simplified value is the remainder.

This computation provided a quick result of \(65\) as the remainder. Understanding this method not only saves time but also deepens your grasp of polynomial behavior and their properties in algebra.