Problem 27 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(9 x^{3}-x+2\right) \div(3 x-1) $$

Step-by-Step Solution

Answer

The quotient is \(9x^2 + 3x\) with a remainder of 2.

Step 1: Setting Up Synthetic Division

Synthetic division is used to divide a polynomial by a binomial of the form \(x - c\). For \(3x - 1\), rewrite it as \(x - \frac{1}{3}\). Thus, \(c = \frac{1}{3}\). Set up the synthetic division by listing the coefficients of the polynomial \(9x^3 + 0x^2 - x + 2\) (note the 0 for the missing \(x^2\) term): \([9, 0, -1, 2]\).

Step 2: Performing Synthetic Division

Bring down the first coefficient (9) as it is. Multiply \(9\) by \(\frac{1}{3}\) to get 3, and add this to the next coefficient (0) to get 3. Multiply 3 by \(\frac{1}{3}\) to get 1, and add this to the next coefficient (-1) to get 0. Multiply 0 by \(\frac{1}{3}\) to get 0, and add this to the last coefficient (2) to get 2.

Step 3: Interpreting the Result

The numbers at the bottom row \([9, 3, 0, 2]\) represent the coefficients of the quotient. The first three numbers \([9, 3, 0]\) are the coefficients of the quotient polynomial obtained, and the last number \(2\) is the remainder. The quotient polynomial is \(9x^2 + 3x + 0\) and the remainder is 2. Therefore, the result is \(9x^2 + 3x + \frac{2}{3x-1}\).

Key Concepts

Polynomial Division

Polynomial division is an essential skill in algebra that allows us to divide two polynomials, gaining a quotient and possibly a remainder. The central idea is closely related to division with integers. This method comes in handy when simplifying expressions or solving polynomial-based equations. When dividing polynomials, especially higher-degree polynomials, traditional long division can be tedious. That's where synthetic division offers a simplified alternative.

• The division involves dividing a polynomial (dividend) like \(9x^3 - x + 2\) by a simpler polynomial called the divisor, such as \(3x - 1\).
• The goal is to express the division result as a quotient polynomial plus a remainder.

This understanding of polynomial division forms the foundation for mastering synthetic division, our next focus.

Remainder Theorem

The Remainder Theorem is a helpful result in algebra that tells us something quite interesting about division. It states that if you divide a polynomial \(f(x)\) by \(x - c\), then the remainder of this division is \(f(c)\).
Applying the theorem in synthetic division means the final number we arrive at is the remainder.

• For the polynomial \(9x^3 - x + 2\), divided by \(3x - 1\), we substitute \(x = \frac{1}{3}\) into the polynomial for calculating the remainder.
• Following the synthetic division process, this remainder was seen as \(2\).

This is a quick way to verify division results or check if a binomial is a factor of the polynomial.

Quotient Polynomial

A key outcome of polynomial division is obtaining the quotient polynomial, which results from the division process. In synthetic division, after completing the operation, the coefficients at the bottom represent the quotient polynomial.
For instance, dividing \(9x^3 - x + 2\) by \(3x - 1\), we acquired \([9, 3, 0]\) as the coefficients.

• This means our quotient polynomial is \(9x^2 + 3x\).
• Each coefficient corresponds to a term in descending order of power, starting from the degree of the dividend minus the degree of the divisor (in this case degree 2).

Understanding the construction of a quotient polynomial assists in reconstructing the expression and verifying our division work.

Cubic Polynomial Division

Dividing cubic polynomials is a specific type of polynomial division where the dividend has a third power.
These are slightly more complex than simpler polynomials because of their higher order. However, synthetic division simplifies this process significantly.

• With the cubic polynomial \(9x^3 - x + 2\), identify the coefficients for synthetic division: \([9, 0, -1, 2]\).
• After performing the division, interpret the coefficients to form the quotient polynomial \(9x^2 + 3x\).
• The remainder is handled separately, as \(\frac{2}{3x-1}\).

Cubic polynomial division through synthetic division eliminates cumbersome calculations and helps in efficiently finding a clear result.