Question

Use synthetic division to divide. Divisor \(x-6\) Dividend $$10 x^{4}-50 x^{3}-800$$

Short answer

The result of the given synthetic division problem is \(10x^{3}-10x^{2}-40=0\), meaning the quotient is \(10x^{3}-10x^{2}-40\) and the remainder is 0.

Step-by-step solution

Setting up the Synthetic Division

First, write the coefficients of the dividend, which are the coefficients of \(10x^{4}\), \(-50x^{3}\), and \(-800\). In synthetic division process, these coefficients are written in a row, inside a bracket-like formation. Also, write the value of \(x\) from the divisor \(x-6\) which is \(6\) outside the bracket-like structure to the left.

Perform Synthetic Division

To perform synthetic division, bring down the first coefficient which is 10 (from \(10x^{4}\)). Multiply the number outside the bracket-like structure (which is 6 from the divisor \(x-6\)) by the number just written under the line (10), and place the result under the next coefficient - in this case, -50. Then, add the numbers in this column (-50 and 60), writing the result (-10) under the line. Repeat this process for the next coefficients.

Write the Final Result

After the whole process of synthetic division, the numbers written under the line represent the coefficients of the quotient polynomial. Starting from one degree less than the original polynomial (which was a fourth degree polynomial), the result, in this case, will be a third degree polynomial. The last number under the line is the remainder, the remaining polynomial is the quotient.

Key concepts

Polynomial Division

Polynomial division is a crucial concept in algebra, allowing you to divide polynomials just like you would with simple numbers. In context, when you divide one polynomial by another, you aim to find the quotient (result) and possibly a remainder. It's akin to dividing something into parts to see how many times one fits into another. In algebra, division can be trickier because you're dealing with variables and coefficients.
Let's start with the basics: the dividend is the polynomial you are dividing, while the divisor is the polynomial you are dividing by. The result you get is the quotient and sometimes, if it does not divide perfectly, you'll be left with a remainder.
The main methods of polynomial division include long division and synthetic division. Synthetic division is particularly handy when the divisor is a linear binomial (i.e., it has a form of \(x - c\)). Its main advantage is that it simplifies the process and makes it quicker compared to long division, especially for students tackling lengthy problems in algebra.

Quotient and Remainder

When you divide an equation and don't get a clean break (like splitting a pie among friends), you end up with a quotient and possibly a remainder. Understanding what these two terms mean is key to division in algebra.

• The **quotient** is the result of the division—the part that indicates how many times the divisor fits inside the dividend.
• The **remainder** is what is left over after you have taken out as many whole units of the divisor as possible. It’s like the leftover slice or the part that doesn’t quite divide evenly.

Synthetic division provides an effective way to quickly find both the quotient and the remainder for polynomials. In the original exercise, after setting up and performing synthetic division, the numbers you get at the bottom line represent the coefficients for your quotient, and the last number is your remainder. The remainder might either be zero or some value representing the leftover portion of the division.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They include numbers, variables, and operations like addition, subtraction, multiplication, and division. Understanding these elements is crucial for working with problems like polynomial division.
In algebra, you often manipulate these expressions to simplify or solve them. When dealing with polynomial division, you are focusing on dividing one algebraic expression by another. It requires knowing how to manage coefficients (numbers in front of the variables) and understanding the rules that govern how variables behave during division.
This particular exercise involves an algebraic expression that’s a polynomial, which means it’s made up of terms with variables raised to powers, such as \(10x^4, -50x^3,\) and \(-800\). Successfully dividing this polynomial using synthetic division depends on understanding how each part of these algebraic expressions interacts with the divisor, resulting in a simplified expression or quotient, and potentially a remainder.