Problem 8 - Precalculus : Building Concepts and Connections

Question

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$2 x^{4}-x^{3}+x^{2}-x ; 2 x+1$$

Step-by-Step Solution

Answer

The quotient is \(x^{3}-\frac{1}{2}x^{2}+\frac{1}{4}x-\frac{1}{8}\) and the remainder is \(\frac{1}{8}\).

Step 1: Setting up the polynomial division

To start with, set up the polynomial division. Write the dividend, which is \(2x^{4}-x^{3}+x^{2}-x\), inside the division bar and the divisor, which is \(2x+1\), to the left of the division bar.

Step 2: Performing the division

Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient. Thus, \(2x^{4}\) divided by \(2x\) gives \(x^{3}\). Now, multiply the divisor \(2x+1\) by \(x^{3}\) to get \(2x^{4}+x^{3}\). Subtract this value from the first two terms of the dividend. The result is \(3x^{2}-x\). Bring down the next term from the dividend, which is \(-x\). Repeat this process until there are no more terms to bring down from the dividend.

Step 3: Obtaining the quotient and remainder

Keep dividing until it's no longer possible to do so. The quotient is the polynomial you get, and the remainder is what's left over that couldn't be divided anymore. In that case, the quotient is \(x^{3}-\frac{1}{2}x^{2}+\frac{1}{4}x-\frac{1}{8}\), and the remainder is \(\frac{1}{8}\).

Key Concepts

Synthetic Division

Synthetic division is a shorthand method for dividing polynomials, primarily used when the divisor is a linear binomial of the form \(x - c\). It's a more efficient process than long division and particularly useful for simplifying calculations. In synthetic division, we focus on the coefficients of the polynomials, making it quicker and easier, especially when working with higher degree polynomials.

Let's walk through the process:

• First, we write down the coefficients of the dividend polynomial. For \(2x^4 - x^3 + x^2 - x\), the coefficients are \(2, -1, 1, -1, 0\). The last zero comes from the missing constant term.
• Next, we write the root of the divisor's zero. Since the divisor is \(2x + 1\), we solve \(2x + 1 = 0\), giving \(x = -\frac{1}{2}\).
• Place this root in a corner position and perform synthetic division by iterating through multiplication and addition of coefficients.

By the end of the synthetic division, the coefficients provide the quotient, and the last value is the remainder. This method doesn't handle non-linear divisors directly, thus should be used when applicable.

Remainder Theorem

The remainder theorem provides a useful way to determine the remainder of a polynomial division without performing the entire division. It's a powerful shortcut in algebra. According to this theorem, if you divide a polynomial \(f(x)\) by a simple binomial \((x-c)\), the remainder of this division is simply \(f(c)\). This means that the remainder equals the value of the polynomial evaluated at \(c\).

For example, using \(2x+1\) as the divisor in the original problem, you find \(c = -\frac{1}{2}\). By substituting \(-\frac{1}{2}\) into \(2x^4 - x^3 + x^2 - x\), you can swiftly find the remainder without extensive computation.

The Remainder Theorem helps confirm if a synthetic division was performed correctly. If after evaluating \(f(c)\), the value obtained matches the remainder from synthetic division, it's a good indication of the verification of the process.

Quotient Polynomial

The quotient polynomial results from dividing one polynomial by another, essentially being the solution to the division minus any remainder. This polynomial tells us how many times the divisor polynomial fits into the dividend polynomial.

When using synthetic division or polynomial long division, the quotient is constructed progressively as each step of the division is completed.

• In the given example, after performing the division, the quotient is found to be \(x^3 - \frac{1}{2}x^2 + \frac{1}{4}x - \frac{1}{8}\).
• These terms result from determining successive terms in the division process until all parts of the dividend are accounted for.

Understanding the quotient polynomial is crucial as it frequently reveals factors or other significant elements of the original polynomial, offering deeper insights into its structure. This is why grasping the concept of quotient polynomial is an important part of mastering polynomial division.