Problem 12 - College Algebra

Question

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{x^{4}-81}{x-3}$$

Step-by-Step Solution

Answer

The quotient \(q(x)\) is \(x^3+3x^2+9x+27\) and the remainder \(r(x)\) is 0.

Step 1: Set Up the Division

Start by setting up the synthetic division. Write the coefficients of the polynomial \(x^{4}-81\) in descending order of power, which are \(1, 0, 0, 0, -81\), and write the roots of the divisor \(x-3\) which is \(3\).

Step 2: Conduct Synthetic Division

Now perform the synthetic division step by step. First, drop down the leading coefficient (1) of the dividend. Then multiply the root (3) by this number and write the result under the second number from the dividend (0). Add the numbers in this new column together to get a new number (3). Repeat this process until all the numbers from the dividend have been used.

Step 3: Write Down the Quotient and Remainder

The numbers obtained in the last row of the synthetic division represent the coefficients of the quotient \(q(x)\), starting from one degree lower than the original polynomial. The remainder \(r(x)\) is the last number obtained in synthetic division process.

Step 4: Finalize the Answer

Write down the final quotient and remainder in their polynomial form. The quotient will be a cubic polynomial since we start from one degree lower than the original, and the remainder will be a constant.

Key Concepts

Synthetic Division

Synthetic division is an abbreviated form of polynomial long division which is particularly useful when dividing a polynomial by a linear factor in the form of \( x - c \). This method is more straightforward and less cumbersome than traditional long division when dealing with polynomials.

To perform synthetic division, you start by identifying the constant \( c \), which is the root of the divisor (in the case of our original exercise, it's 3, because we're dividing by \( x - 3 \)). Next, the coefficients of the polynomial are listed in order, inclusive of zeros for any missing terms.

Here's a step-by-step breakdown:

• Write down the coefficients of \( x^4 - 81 \), which are 1, 0, 0, 0, -81.
• The root of \( x - 3 \) is 3, which you'll write to the side of the coefficients.
• Bring down the first coefficient (1).
• Multiply the root (3) by this coefficient and place the result under the second coefficient (0).
• Continue the process by adding vertically and multiplying by the root, until you've worked through all coefficients.

The last row of numbers represents the coefficients of the quotient, with the last number being the remainder.

It's effective, quick, and serves as an excellent tool for confirming roots of polynomial equations or simplifying the division of polynomials.

Dividing Polynomials

Dividing polynomials is a process similar to long division in arithmetic, where one polynomial (the dividend) is divided by another polynomial (the divisor), resulting in a quotient and sometimes a remainder. The goal is to find how many times the divisor can 'fit into' the dividend.

When the divisor is a linear polynomial like \( x - c \), synthetic division is a favored method. However, for divisors of higher degree, polynomial long division is generally used.

To divide polynomials using long division, you would:

• Write the dividend and divisor in the standard long division symbol.
• Determine how many times the leading term of the divisor can divide into the leading term of the dividend.
• Subtract the product of the divisor and the determined number from the dividend.
• Repeat the process with the new, lower degree polynomial you get after subtraction.

You continue in this manner until the degree of the remainder is less than the degree of the divisor or until the remainder is zero.

Mastering this technique is crucial for anyone studying algebra, as it is used in simplifying algebraic expressions and solving polynomial equations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific value. These expressions can range from the simple, such as \( 2x + 3 \), to the complex, like the polynomial \( x^4 - 81 \) featured in our exercise. Algebraic expressions are the building blocks of algebra and are manipulated using various operations including addition, subtraction, multiplication, division, and exponentiation.

An important aspect of algebraic expressions is the ability to simplify and factor them. Simplification can involve combining like terms or reducing fractions, while factoring involves rewriting expressions as a product of their factors. Both skills are essential for solving equations and analyzing functions.

By understanding the structure of algebraic expressions and learning how to manipulate them through operations such as dividing polynomials using polynomial long division or synthetic division, students can solve a wide array of mathematical problems and deepen their comprehension of algebraic concepts.