Problem 325 - Precalculus

Question

Use synthetic division to find the quotient $$\left(x^{3}-3 x+2\right) \div(x+2)$$

Step-by-Step Solution

Answer

The quotient is \( x^2 - 2x + 1 \).

Step 1: Write Down the Dividend and Divisor

The dividend is the polynomial we want to divide, which is \( x^3 - 3x + 2 \). Make sure all terms are present by writing it as \( x^3 + 0x^2 - 3x + 2 \). The divisor is \( x + 2 \), and we will use the root of this expression, \( -2 \), for synthetic division.

Step 2: Set Up Synthetic Division

Write \( -2 \) on the left side. Write the coefficients of the dividend \( 1, 0, -3, 2 \) in a row to the right of it. This sets up our synthetic division chart.

Step 3: Bring Down the Leading Coefficient

Bring down the first coefficient \( 1 \) directly below the line. This will serve as the beginning of the quotient.

Step 4: Multiply and Add Down

Multiply the number just brought down (\( 1 \)) by \( -2 \), the divisor's root. Write the result (\(-2\)) under the next coefficient (\( 0 \)). Add these two numbers to get \(-2\) and write this below the line.

Step 5: Repeat the Multiply and Add Process

Multiply \(-2\) by \(-2\) to get \( 4 \). Write it under the next coefficient \(-3\). Add these numbers to get \( 1 \) and write this below the line. Next, repeat: multiply \( 1 \) by \(-2\) to get \(-2\) and add to 2, resulting in \( 0 \).

Step 6: Interpret the Result

The row of numbers below the line, from left to right, are the coefficients of the quotient polynomial. Since the remainder is \( 0 \), the original polynomial is exactly divisible. The quotient is \( x^2 - 2x + 1 \).

Key Concepts

Polynomial Division

Polynomial division is similar to the long division process used with numbers. Instead of dividing numbers, we divide polynomials. This method helps solve complex polynomial equations and simplifies expressions. To divide two polynomials, we separate the dividend (the polynomial we want to divide) and the divisor (the polynomial we divide by). In our exercise, the dividend is \( x^3 - 3x + 2 \) and the divisor is \( x + 2 \).
The goal is to find out how many times the divisor fits into the dividend, a process which leads us to the quotient. To make the division simpler and more organized, synthetic division is often used when the divisor is linear (having a degree of 1).
Synthetic division is a shortcut method that skips the variables and focuses on the coefficients, making it faster and easier. The method also efficiently reveals the remainder, if any, left after division.

Quotient Polynomial

The quotient polynomial is the result we get in polynomial division when the divisor is successfully divided into the dividend. For our exercise, we specifically apply synthetic division, a simplified method.
We began with the polynomial \( x^3 + 0x^2 - 3x + 2 \) and divided it by \( x + 2 \).

• The coefficients are lined up: \( 1, 0, -3, 2 \).
• We use the root of the divisor, \( -2 \), to perform the division.
• The process involves bringing down the leading coefficient and continuously multiplying by \( -2 \), while adding down the column to get the successive coefficients.

After completing the process, the line of numbers revealed the coefficients of the quotient polynomial: \( x^2 - 2x + 1 \). This quotient shows how the divisor fits into the dividend perfectly, leaving no remainder.

Remainder in Division

In polynomial division, the remainder is the leftover when the divisor does not completely divide the dividend. It tells us how much extra or deficit there is after fitting the divisor into the dividend.
The importance of the remainder is similar to that in numerical division.

• If the remainder is zero, the division is exact, meaning the divisor perfectly divides the dividend with no extras.
• If there is a remainder other than zero, this value is what remains after the division process.

In our exercise, we observed the remainder by looking at the last position of the synthetic division output. Here, the remainder was zero. This indicates that \( x^3 - 3x + 2 \) is completely divisible by \( x + 2 \), confirming the given quotient polynomial is the exact result without leftover terms.