Question

Use long division to divide. Divisor \(x-2\) Dividend $$x^{3}-x^{2}+2 x-8$$

Short answer

The result of the division is \(x^2 + 3x + 8\).

Step-by-step solution

Set Up the Division

First, it's necessary to set up the problem like a traditional long division problem, with the dividend inside the artificial division bar and the divisor outside to the left. So, write \(x^{3}-x^{2}+2x-8\) inside the bar, and \(x-2\) on the outside.

Divide the Highest Degree Terms

Next, divide the highest degree term of the dividend \(x^3\) by the highest degree term of the divisor \(x\). In this case, this yields \(x^2\). Write this answer on top of the division bar.

Multiply the Entire Divisor by the Answer from Step 2

Now, multiply the entire divisor \(x-2\) by the answer from step 2, \(x^2\). This results in \(x^3-2x^2\). Write these terms beneath their corresponding terms in the dividend, draw a line beneath them, and subtract.

Carry down the Next Term

With the subtraction in step 3, the \(x^3\) terms cancel each other, leaving \(x^2 - (-2x^2) = 3x^2\). Carry down the next term from the dividend, which is \(+2x\), resulting in \(3x^2 + 2x\).

Repeat the Process

Repeat steps 2 to 4 with the new dividend \(3x^2+2x\). Dividing \(3x^2\) by \(x\) gives \(3x\), multiplying \(x-2\) by \(3x\) gives \(3x^2-6x\). Subtract this from \(3x^2+2x\) leaves \(8x\). Carry down next term from the dividend \(-8\), so it becomes \(8x - 8\).

Repeat Step 2 to 5 One Last Time

One more time, divide \(8x\) by \(x\) gives \(8\). Multiply \(x-2\) by \(8\) gives \(8x-16\), subtracting this from \(8x-8\) gives \(8\), which is the remainder. Thus, the entire division process is finished.

Write the Final Answer

The answer consists of the terms written on the top of the division bar and any remainder, written as fractional part. In this case, the quotient of the division is \(x^2 + 3x + 8\), with no remainder.

Key concepts

Polynomial Division

In algebra, division of polynomials is similar to the process of long division with numbers. When dividing polynomials, we aim to find how many times the divisor can "fit" into the dividend. The goal is to express a given polynomial as the product of the divisor and the quotient, plus a remainder, if there is any. This process can simplify complex algebraic expressions or solve polynomial equations.
Understanding polynomial division involves learning how to work with terms and coefficients. It also requires following a systematic procedure to ensure accuracy. This ensures we manage every degree of terms correctly, from the highest down to the lowest, just like in long division with real numbers.

Algebraic Expressions

An algebraic expression consists of constants, variables, and operations, such as addition, subtraction, multiplication, and division. They play an essential role in mathematics, as they describe mathematical relationships and problem-solving.
In the context of polynomial division, algebraic expressions need careful manipulation. When performing operations on these expressions, it's crucial to maintain the integrity of each unit by following rules of arithmetic and algebra, like combining like terms and distributing multiplication over addition and subtraction.

Dividing Polynomials

Dividing polynomials is a valuable skill in algebra that allows simplification of expressions or solving equations. This process typically involves steps such as setting up the division, dividing the highest-degree terms, multiplying and subtracting, and repeating until complete.

• Set up the long division with the dividend and divisor appropriately.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result from the division, and subtract from the dividend.

We continue this process, bringing down terms as needed, until we cannot divide any further.

Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, specifically when the divisor is linear, like in our exercise ( x - 2"). It's a more efficient alternative to long division when applicable. Unlike long division, synthetic division uses only the coefficients of the polynomial.
Here's a basic rundown of how synthetic division works:

• List the coefficients of the dividend.
• Use the root of the divisor (the value that makes it zero) as the divisor.
• Perform the synthetic division algebraically to find the quotient and remainder.

Synthetic division is especially useful because it often reduces the amount of computation required, helping simplify problems quicker.