Problem 387 - Precalculus

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(4 x^{3}+5 x^{2}-2 x+7\right) \div(x+2) $$

Step-by-Step Solution

Answer

The remainder is -1.

Step 1: Understanding the Problem

We want to find the remainder when the polynomial \( 4x^3 + 5x^2 - 2x + 7 \) is divided by \( x + 2 \). According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x-c \) is given by \( f(c) \).

Step 2: Identifying the Value of c

In this problem, the polynomial is divided by \( x + 2 \). Here, \( x + 2 = x - (-2) \), so \( c = -2 \).

Step 3: Substitute the Value into the Polynomial

We substitute \( c = -2 \) into the polynomial \( f(x) = 4x^3 + 5x^2 - 2x + 7 \). Evaluate \( f(-2) \).

Step 4: Calculate Each Term Separately

First, calculate each term: - \( 4(-2)^3 = 4(-8) = -32 \) - \( 5(-2)^2 = 5(4) = 20 \) - \( -2(-2) = 4 \)- The constant term is \( 7 \).

Step 5: Compute the Remainder

Sum the values of each term: \( -32 + 20 + 4 + 7 = -1 \).Thus, the remainder is \(-1\).

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers. When dividing one polynomial by another, our aim is to find the quotient and remainder. If we divide polynomial \( f(x) \) by another polynomial \( d(x) \), we want to write it in the form:

• \( f(x) = d(x) \cdot q(x) + r(x) \)

where \( q(x) \) is the quotient and \( r(x) \) is the remainder. The degree of \( r(x) \) should always be less than the degree of \( d(x) \).
In many cases, it's not necessary to compute the quotient to determine the remainder. The Remainder Theorem makes this process much easier by providing a shortcut. By using this theorem, we can directly find the remainder without performing lengthy division. We simply evaluate \( f(x) \) at a specific value related to the divisor, which is detailed more in the next sections.

Evaluating Polynomials

Evaluating a polynomial involves calculating its value at a particular point. This is especially useful in the context of the Remainder Theorem.The Remainder Theorem states that for a polynomial \( f(x) \) divided by \( x-c \), the remainder is simply \( f(c) \).To evaluate the polynomial \( f(x) = 4x^3 + 5x^2 - 2x + 7 \) at \( x = -2 \), we replace every instance of \( x \) with \( -2 \):

• \( 4(-2)^3 = -32 \)
• \( 5(-2)^2 = 20 \)
• \( -2(-2) = 4 \)
• The constant term remains \( 7 \).

When we add all these results together, we get \( -32 + 20 + 4 + 7 = -1 \). Hence, the remainder of this division is \( -1 \).
This method is simple and avoids the need for complex polynomial division, which makes it highly efficient especially for polynomials with higher degrees.

Synthetic Division

Synthetic division is a streamlined method of polynomial division, particularly useful when dividing by a binomial of the form \( x-c \). Compared to traditional long division, synthetic division requires fewer steps and is easier to perform by hand. Here's a quick overview of how it works:

• Write down the coefficients of the polynomial.
• Use the value \( c \) (from \( x-c \)) for the division process.
• Bring down the leading coefficient to start the process.
• Multiply and add in succession across the row of coefficients.
• The last number you get is the remainder.

In our example, dividing \( 4x^3 + 5x^2 - 2x + 7 \) by \( x+2 \) (thus \( c = -2 \)), synthetic division simplifies the task. This technique gives the same remainder as obtained using the Remainder Theorem, affirming its efficiency and effectiveness for quick checks.