Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Use synthetic division to divide. Divisor \(x+3\) Dividend $$2 x^{5}$$

Step-by-Step Solution

Answer

In this scenario, synthetic division of \(2x^5\) by \(x+3\) results in a quotient of \(2x^4\) with no remainder.

Step 1: Setup

To set up the synthetic division, an understanding of the coefficients of every term in the dividend polynomial is needed. The given dividend is \(2x^5\), indicating that all the coefficients of \(x^4\), \(x^3\), \(x^2\), \(x\), and the constant term are zero. The divisor is \(x + 3\). Be aware that when setting up synthetic division, there is a need to change the sign of the term added or subtracted from x in the divisor. Therefore, the synthetic division setup would involve -3 (not +3) from the divisor and the coefficients 2, 0, 0, 0, 0, 0 from the dividend.

Step 2: Synthetic Division

Start synthetic division by dropping down the leading coefficient (which is 2 in this case). Multiply the number you just wrote by the value you're dividing by (-3 in this case). Write this product under the next coefficient and add it up. Repeat this process until all terms have been worked through.

Step 3: Define the Result

After carrying out the synthetic division, the result becomes the coefficients of a polynomial with degree less than the dividend. If after dividing, there's a nonzero value in the last row, this is your remainder and the polynomial would be in the format: quotient(x) + remainder / (divisor). When there's no remainder (which can be expected from this problem), the result would just be the polynomial from the coefficients gained from the synthetic division.

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another. It serves a similar purpose as regular division but is specifically tailored for polynomials. The key goal is to break down complex polynomial expressions into simpler parts. There are two primary methods for polynomial division:

• Long Division
• Synthetic Division

Both methods aim to achieve the same goal, but synthetic division is often preferred when the divisor is of the form \(x + c\), due to its simplicity and efficiency.

Coefficients

Coefficients are the numerical factors in terms of a polynomial. For example, in the term \(2x^5\), the coefficient is 2. These numbers are crucial when executing synthetic division. When setting up synthetic division, all coefficients of the polynomial need to be accounted for, which includes placeholder zeros for any missing terms. In the exercise given, the polynomial \(2x^5\) demonstrates this with the coefficients being:

• 2 for \(x^5\)
• 0 for \(x^4\), \(x^3\), \(x^2\), \(x^1\), and the constant term

This highlights the need to include zeros to ensure that all degree levels are represented in the process.

Remainder Theorem

The Remainder Theorem is a useful concept in polynomial mathematics. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder of this division is simply \(f(c)\). This theorem provides a quick way to calculate remainders without performing full-fledged division. In synthetic division, the outcome of division includes a remainder, which can be verified using this theorem. In the provided exercise, since the divisor is transformed to \(-3\) from \(x+3\), checking \(f(-3)\) brings a deeper understanding of any remainder that might result, although, in this case, the remainder is zero.

Polynomial Functions

Polynomial functions are expressions composed of variables raised to whole number exponents with coefficients. These functions form the foundation of algebra and calculus due to their flexibility and power. They are commonly expressed in standard form, which lists terms in decreasing order of degree. For instance:

• \(2x^5\) is a simple polynomial with a single term.
• Complex polynomials might look like \(3x^6 + 2x^3 - x + 5\).

The degree of a polynomial function is the highest degree of its terms. Understanding polynomial functions, like recognizing their structure and characteristics, is essential when engaging with synthetic division, as it involves manipulating these expressions into divisors and dividends.