Problem 33 - Algebra and Trigonometry

Question

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{5}+3 x^{3}-6}{x-1} $$

Step-by-Step Solution

Answer

Quotient: \(x^4 + x^3 + 4x^2 + 4x + 4\); Remainder: \(-2\).

Step 1: Set Up Synthetic Division

To begin synthetic division, write down the coefficients of the polynomial you are dividing, omitting any missing terms. For the given polynomial \(x^5 + 0x^4 + 3x^3 + 0x^2 + 0x - 6\), the coefficients are \(1, 0, 3, 0, 0, -6\). The divisor \(x - 1\) means we use \(1\) as the divisor.

Step 2: Begin Synthetic Division

Draw a horizontal line with a box above it. Place the divisor \(1\) to the left of the vertical line. Write the coefficients \(1, 0, 3, 0, 0, -6\) in a row to the right of the line. Bring down the first coefficient \(1\) below the line:

Step 3: Perform Synthetic Division

Multiply the divisor \(1\) by the number just brought down (\(1\)), and write the result \(1\) beneath the next coefficient (\(0\)). Add, resulting in \(0 + 1 = 1\). Continue this process:1. Multiply \(1\) by \(1\) and add to the next coefficient \(3\), resulting in \(4\).2. Multiply \(1\) by \(4\) and add to \(0\), resulting in \(4\).3. Multiply \(1\) by \(4\) and add to \(0\), resulting in \(4\).4. Multiply \(1\) by \(4\) and add to \(-6\), resulting in \(-2\).

Step 4: Identify Quotient and Remainder

The "bottom row" numbers from synthetic division are the coefficients of the quotient: \(1, 1, 4, 4, 4\), representing \(x^4 + x^3 + 4x^2 + 4x + 4\). The last number is the remainder, \(-2\):\[\frac{x^5 + 3x^3 - 6}{x-1} = x^4 + x^3 + 4x^2 + 4x + 4 - \frac{2}{x-1}.\]

Key Concepts

Polynomial Division

Polynomial division is a way to divide a polynomial by another polynomial, usually of lower degree. It's similar to long division with numbers but involves variables. In this case, we are dividing the polynomial \(x^5 + 3x^3 - 6\) by \(x - 1\).

The goal is to find the quotient, which is a new polynomial, and the remainder, which is a smaller degree term that the divisor cannot further divide. Here's why polynomial division is useful:

• It simplifies polynomials, turning complex expressions into easier ones.
• It helps in factoring polynomials, particularly when checking for roots.
• It aids in understanding the behavior of polynomial functions.

Understanding polynomial division is fundamental, not only for solving equations but also for calculus and higher mathematics. Synthetic division, a shortcut method, allows for quicker division when the divisor is of the form \(x - c\). It saves time especially when handling higher degree polynomials.

Remainder Theorem

The Remainder Theorem is a helpful concept that connects polynomial division and roots of polynomials. It states that when a polynomial \(P(x)\) is divided by \(x - c\), the remainder of this division is simply \(P(c)\). In our example, the polynomial \(x^5 + 3x^3 - 6\) is divided by \(x - 1\), and the remainder is \(-2\).

Let's break down why this is vital:

• If the remainder is \(0\), it implies \(x = c\) is a root of the polynomial.
• Remainder Theorem provides a quick way to check potential roots.
• It can be used to verify synthetic division results.

For instance, if you substituted \(x = 1\) in our polynomial and found that it equals the remainder \(-2\), it confirms correctness. Mastering this theorem helps in efficient problem-solving in polynomial-related questions.

Quotient in Division

The quotient in polynomial division is the polynomial result you get after dividing one polynomial by another. In our exercise, using synthetic division, we found that the quotient is \(x^4 + x^3 + 4x^2 + 4x + 4\). This happens after completing synthetic division, separating the initial polynomial into a quotient and remainder.

Remember these points about the quotient in division:

• It reduces the degree of the original polynomial by the degree of the divisor.
• Helps in factoring and simplifying expressions.
• Gives insight into polynomial behavior and characteristics.

Getting the quotient correct ensures the rest of your calculations regarding the polynomial, such as finding zeros or graphing, are also spot-on. Practices like synthetic division make obtaining the quotient quicker and less prone to calculation errors.