Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Use synthetic division to divide. Divisor \(x+2\) Dividend $$6 x^{4}-15 x^{3}-11 x$$

Step-by-Step Solution

Answer

The result of the synthetic division is \(6x^3 -27x^2 + 54x -108 + \frac{216}{x+2}\).

Step 1: Identify the coefficients

First, arrange the polynomial by degree in descending order which is \(6x^4-15x^3+0x^2 -11x+0\). Identify and list the coefficients of dividend and the root of divisor. The coefficients of the dividend are 6, -15, 0, -11, and 0 respectively since there are no terms in \(x^2\) or the constant term so we use 0. The root of the divisor \(x+2\) is -2.

Step 2: Setup the synthetic division

Set up the synthetic division by writing the root on the left, then draw a L-shaped line. Write the coefficients along the top inside the 'L' shape.

Step 3: Perform the synthetic division

To start synthetic division, bring down the leading coefficient (which is 6 in this case). Multiply the root by the number you've written below the division line, then write the result under the next coefficient and add them, repeat this process until you've worked with all coefficients.

Step 4: Write out the result

The result of synthetic division gives coefficients of the quotient and remainder. The last number on the bottom row is the remainder and the other numbers are coefficients of the quotient. The power of the variable for the first term of the quotient is one less than the power of the variable in the first term of the original polynomial.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used in mathematics to divide one polynomial by another. It is similar to the long division process used with numbers but involves variables raised to various powers.

Let's imagine you are dividing 6x^4 by x+2. You would start by dividing the highest degree term in the divisor (x) into the highest degree term in the dividend (6x^4), which gives you 6x^3. This result is then multiplied by the entire divisor and subtracted from the dividend, which gives you a new polynomial. This process is repeated until the degree of the remainder is less than the degree of the divisor.

The result consists of a quotient and often a remainder. The quotient is the polynomial resulting from the division without the remainder. The process requires careful attention to coefficients and exponents and can be lengthy, which is why synthetic division offers a more streamlined alternative when applicable.

Dividing Polynomials

Dividing polynomials is an essential part of algebra that can be solved using either polynomial long division or synthetic division, depending on the nature of the polynomials involved.

When dividing polynomials, it's important to first write both the dividend and the divisor in descending power order. If a polynomial is missing a term, you insert a placeholder with a zero coefficient for that term's degree. For example, in the polynomial 6x^4 - 15x^3 - 11x, you would add 0x^2 and 0 for the missing x^2 and constant term.

Understanding the relationship between terms and their coefficients, as well as how to manipulate them during the division process, is crucial. After dividing, the final result will give you not only the quotient but may also provide a remainder, and this is applicable to both long division and synthetic division.

Algebraic Division Techniques

Algebraic division techniques, such as polynomial long division and synthetic division, are strategies used to solve algebraic expressions where division of polynomials is required.

While long division is more universal and can handle more complex polynomials, synthetic division is a shortcut method that works only when dividing by a linear divisor of the form x - c. It simplifies the process by focusing on the coefficients and the constant term of the linear divisor.

The steps involved in these techniques require a firm grasp of basic algebraic manipulation, including adding, subtracting, and distributing polynomials. By practicing these methods, students can tackle a wide range of problems more efficiently, gaining a deeper understanding of polynomials and their behavior during division.

Solving Algebraic Expressions

Solving algebraic expressions often involves a variety of techniques, including division of polynomials.

Whether you're simplifying expressions, finding factors, or solving equations, the ability to divide polynomials accurately is vital. In context, after completing a division using either long division or synthetic division, you may arrive at a simpler expression which can then be used to further analyze or solve problems. For instance, finding zeros of a polynomial or simplifying a rational expression.

Thoroughly understanding the steps in each division technique ensures that the process of solving these algebraic expressions becomes less daunting and more approachable. Dividing polynomials is not merely a mathematical exercise but also a foundational tool for working with and understanding higher-level algebraic concepts.