Question

Use synthetic division to divide. Divisor \(x-6\) Dividend $$180 x-x^{4}$$

Short answer

The resulting polynomial from the synthetic division of \(-x^4 + 180x\) by \(x-6\) is given as \(f(x) = -x^3 + 6x^2 - 36x + 216\).

Step-by-step solution

Rewrite the divisor and the dividend

First, rewrite the divisor as \(x = 6\), because synthetic division is defined in terms of the zeroes, not the factors. Then, write the dividend in descending order of exponent, including terms with zero coefficients for any missing powers. Here, the dividend becomes \(-x^4 + 0x^3 + 0x^2 + 180x + 0\).

Set up synthetic division

Arrange the coefficients of the dividend (-1, 0, 0, 180, 0) into a row, and put the zero of the divisor (6) in a box to the left. This creates the framework for synthetic division.

Perform synthetic division

To perform synthetic division, follow these steps: Bring down the first coefficient (-1). Multiply the boxed number (6) by the number just written, and write the result under the second coefficient. Add the second coefficient to this number and write the result beneath. Continue this process until all coefficients have been used.

Interpret the result

The numbers obtained at the bottom of the synthetic division represent the coefficients of the quotient polynomial. The last number is the remainder. The degree of the quotient is one less than the degree of the dividend. Here, the output of the synthetic division is going to be a third degree polynomial because the given dividend is a fourth degree polynomial.

Key concepts

Polynomial Long Division

Polynomial long division is a technique used to divide one polynomial by another, yielding a quotient and sometimes a remainder, much like long division with integers. It is an important tool in algebra that helps with simplifying complex expressions and solving high-degree polynomial equations.

Here's a glimpse into the process:

• Like traditional long division, you write down the dividend and divisor.
• The leading term of the divisor is used to divide the leading term of the dividend.
• The result is multiplied by the entire divisor and subtracted from the dividend, creating a new, typically lower-degree polynomial.
• This process is repeated until the degree of the remainder is less than the degree of the divisor or until a zero remainder is obtained.

Polynomial long division is especially useful when the divisor is a simple binomial, as is often the case in algebraic problems.

Dividing Polynomials

Dividing polynomials is an operation aiming to break down complex polynomial expressions into simpler ones. This includes breaking down a polynomial into factors or separating it into a quotient and remainder. There are two common methods used to divide polynomials: synthetic division, which is a shortcut applicable when the divisor is a first-degree binomial, and polynomial long division, which can be used for divisors of any degree.

Both methods require that you:

• Understand the relationship between the dividend (the polynomial to be divided) and the divisor (the polynomial by which you divide).
• Recognize the terms of the polynomials and arrange them in descending order of their exponents.
• Apply the division steps carefully to avoid mistakes in signs and coefficients.

Successfully dividing polynomials simplifies many algebraic tasks, from finding roots to integrating rational functions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They are the building blocks of algebra and serve as a universal language to represent and solve problems in a symbolic form. Polynomials are a type of algebraic expression that consist of terms, which are products of numbers (coefficients) and variables raised to non-negative integer powers.

Understanding algebraic expressions is crucial for:

• Modeling real-world situations and finding unknown values.
• Performing operations such as addition, subtraction, multiplication, and division.
• Factoring, expanding, simplifying, and transforming expressions to solve equations or inequalities.

With each algebraic operation, the goal is to manipulate the expressions to achieve a desired form, often to isolate a variable or to express the solution in the simplest terms possible.