Question

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. Use synthetic division to find the quotient and remainder when \(2 x^{4}-7 x^{3}+7 x+6\) is divided by \(x-3\).

Short answer

The quotient is \[2x^3 - x^2 - 3x - 2\] and the remainder is 0.

Step-by-step solution

- Set Up Synthetic Division

First, identify the coefficients of the polynomial. For the polynomial \(2x^4 - 7x^3 + 7x + 6\), the coefficients are \2, -7, 0, 7, 6\. The zero of the divisor \(x - 3\) is \3\.

- Write the Coefficients and Zero

Set up the synthetic division by writing the zero \3\ on the left, and the coefficients \[2, -7, 0, 7, 6\] in a row on the right.

- Perform Synthetic Division

1. Drop down the first coefficient \[2\].2. Multiply \[2\] by the zero \[3\] and write the result \[6\] under the next coefficient.3. Add \[-7\] and \[6\] to get \[-1\].4. Repeat the process: - Multiply \[-1\] by \[3\] to get \[-3\]. - Add \[0\] and \[-3\] to get \[-3\]. - Multiply \[-3\] by \[3\] to get \[-9\]. - Add \[7\] and \[-9\] to get \[-2\]. - Multiply \[-2\] by \[3\] to get \[-6\]. - Add \[6\] and \[-6\] to get \[0\].

- Read the Quotient and Remainder

The resulting row of coefficients, \[2, -1, -3, -2\], represents the quotient polynomial \[2x^3 - x^2 - 3x - 2\]. The last value \[0\] is the remainder.

Key concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial. It is similar to long division for numbers. You can use either long division or synthetic division to achieve this. While long division can be more intuitive for some, synthetic division is often quicker and requires less writing.
When performing polynomial division, you're essentially breaking down your dividend polynomial (the one you're dividing) into products and remainders using the divisor polynomial.

Quotient and Remainder

When dividing polynomials, it's important to understand the quotient and remainder. The quotient is the result you get after performing the division, excluding any remaining part. The remainder is what is left over after the division is complete.
For example, in the exercise, when dividing the polynomial \(2x^4 - 7x^3 + 7x + 6\) by \(x - 3\), we performed synthetic division. The quotient turned out to be \(2x^3 - x^2 - 3x - 2\), and the remainder was \(0\).
The remainder can be zero, which means the divisor is a factor of the dividend polynomial, or it can be a non-zero value indicating that the division isn't perfect.

Coefficients

Coefficients are the numerical part of the terms in a polynomial. They play a crucial role in both the setup and execution of polynomial division. For instance, in the polynomial \(2x^4 - 7x^3 + 7x + 6\), the coefficients are \[2, -7, 0, 7, 6\].
In synthetic division, these coefficients are used directly to simplify division. Knowing how to identify and work with them is essential.
In the given exercise, these coefficients are first aligned along with the zero of the divisor. Through systematic multiplication and addition of these coefficients with the zero, we decode the quotient and remainder quickly.