Problem 31 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{4}+2 x^{3}-3 x^{2}+2 x+6\right) \div(x+3) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 - x^2 + 2\) with no remainder.

Step 1: Setup the Synthetic Division

Write the coefficients of the dividend polynomial \(x^4 + 2x^3 - 3x^2 + 2x + 6\) in descending order of power: 1, 2, -3, 2, 6. The divisor is \((x + 3)\), and we use \(-3\) for the synthetic division (since \(x + 3 = 0\) implies \(x = -3\)).

Step 2: Bring Down the First Coefficient

Bring down the leading coefficient from the dividend, which is 1. This will be the first coefficient of the quotient.

Step 3: Multiply and Add

Multiply the number you brought down (1) by the \(-3\) you got from the divisor. Write the result underneath the next coefficient (2). Add these two numbers to get the next number of the quotient: \[1, (2 - 3 = -1)\].

Step 4: Repeat the Process

Continue the multiply and add process for the remainder of the coefficients:- Multiply \(-1\) by \(-3\), add to \(-3\) to get 0.- Multiply 0 by \(-3\), add to 2 to get 2.- Multiply 2 by \(-3\), add to 6 to get 0.

Step 5: Write the Quotient and Remainder

The coefficients you obtained (1, -1, 0, 2) form the quotient. Therefore, the quotient polynomial is \(x^3 - x^2 + 2\). The last number obtained (remainder) is 0, indicating there is no remainder.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial of a lower degree. It is similar to long division with numbers, but in this case, variables are involved. This comes in handy when we need to simplify polynomials or find roots.

• When performing polynomial division, ensure the dividend is written in standard form, where its terms are arranged from highest to lowest degree.
• The divisor, another polynomial, should also be in standard form. The divisor cannot be of a higher degree than the dividend.
• There are two common techniques for polynomial division: long division and synthetic division. Synthetic division is often preferred for its simplicity when the divisor is of the form \(x - c\).

When using synthetic division, we focus on the coefficients rather than the full polynomial terms, which simplifies calculations. This method mainly applies to polynomials with linear divisors.

Remainder Theorem

The remainder theorem is a useful property in polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder is equal to \(f(c)\).

• This theorem allows us to quickly evaluate polynomials at specific points.
• If the remainder is zero, \(c\) is a root of the polynomial, meaning \(x - c\) is a factor.
• In our specific exercise, after performing synthetic division, the remainder is zero, indicating \(x + 3\) is a factor of the dividend polynomial.

Utilizing the remainder theorem after division can also help in verifying calculations and understanding the relationships between polynomials and their roots or factors.

Divisor and Dividend

In the context of polynomial division, it is essential to understand what the divisor and dividend are.

• The dividend is the polynomial that is being divided. In our exercise, it is \(x^4 + 2x^3 - 3x^2 + 2x + 6\).
• The divisor is the polynomial that divides the dividend. Here, it is \(x + 3\).

Think of the divisor and dividend as the participants in division, similar to numbers in arithmetic division. The dividend is the larger number (or polynomial), while the divisor is what we "divide by." Understanding their roles is crucial for setting up and performing synthetic division correctly.

Quotient Polynomial

The quotient polynomial is the result of dividing one polynomial by another, excluding any remainder.

• It represents how many times the divisor can "fit" into the dividend when dividing polynomials.
• In our worked example, the synthetic division yielded a quotient of \(x^3 - x^2 + 2\).
• The coefficients obtained during synthetic division form this quotient.

This simplified form can be useful in various applications, including solving equations and analyzing the behavior of functions. It is essential to interpret each term of the quotient polynomial correctly, as it directly represents the division relationship between the dividend and the divisor.