Question

Use long division to divide. Divisor \(x^{2}-4\) Dividend $$x^{4}+2 x^{3}-3 x^{2}-8 x-4$$

Short answer

The result of the division is \(x^{2}+2x+1-4/(x^{2}-4)\).

Step-by-step solution

Setting up the long division

First, write the problem in long division format. Write \(x^{4}+2x^{3}-3x^{2}-8x-4\) inside the long division symbol and \(x^{2}-4\) outside it.

Divide the leading term

Divide the leading term of the dividend, \(x^{4}\), by the leading term of the divisor, \(x^{2}\), which gives \(x^{2}\). Write this above the long division line. Multiply the divisor \(x^{2}-4\) by \(x^{2}\), the result is \(x^{4}-4x^{2}\) and write this underneath \(x^{4}+2x^{3}-3x^{2}\).

Subtract and bring down

Subtract the line from the previous step \(x^{4}-4x^{2}\) from \(x^{4}+2x^{3}-3x^{2}\) to obtain a new dividend of \(2x^{3}+x^{2}-8x-4\), remembering to change the sign of each term when subtracting. Then bring down the next term from the original dividend, which is \(-8x\).

Divide, Multiply and Subtract Again

Repeat the process. Divide the leading term of the new dividend \(2x^{3}\) by the leading term of the divisor \(x^{2}\) giving \(2x\), write this on top of the division line next to \(x^{2}\). Multiply the divisor \(x^{2}-4\) by \(2x\) giving \(2x^{3}-8x\), write this underneath the new dividend. Subtract the line to obtain a new dividend of \(x^{2}\).

Repeat Until Degree of the Remainder is Less Than the Degree of the Divisor

Repeat the process again by dividing \(x^{2}\) by \(x^{2}\) which is \(1\), multiply \(x^{2}-4\) by \(1\) which gives \(x^{2}-4\), subtract this to obtain \(-4\) which now has a lower degree than \(x^{2}-4\). So, \(-4\) is the remainder and the final result of the division is \(x^{2}+2x+1\).

Write the final result

Now, having finished the division, write the final answer. This is the quotient obtained from above the line \(x^{2}+2x+1\) plus the remainder divided by the divisor \(-4/(x^{2}-4)\). So, the final answer is \(x^{2}+2x+1-4/(x^{2}-4)\).

Key concepts

Polynomial Division

Polynomial division is a method used to divide algebraic expressions, just like long division with numbers. The process we follow here is similar, but we work with polynomials instead of numerical digits. To divide polynomials, the goal is to express the dividend as a product of the divisor and another polynomial, plus a remainder if necessary.

• We start by setting up the division just like in numerical long division. The dividend, the polynomial you want to divide, is placed under the long division symbol.
• The divisor, the polynomial you’re dividing by, is placed outside.
• The aim is to find a quotient that when multiplied by the divisor will produce a term similar to the leading term of the dividend.

In this case, we began by dividing the highest degree term of the dividend by the highest degree term of the divisor, and this guided us through solving the rest of the division process.

Remainder Theorem

The remainder theorem is a useful tool in algebra when dealing with polynomial division. It states that the remainder of the division of a polynomial by a linear divisor is equal to the value of the polynomial evaluated at the root of the divisor. Although our exercise involves a quadratic divisor, understanding the theorem is still beneficial.

• For a linear divisor (like \(x - c\)), the remainder when dividing \(P(x)\) by \(x - c\) is \(P(c)\).
• In cases with higher degree divisors, like \(x^2 - 4\), the remainder might need to be lower-degree, as it was in this exercise.

Knowing the Remainder Theorem helps us check our division work. If the remainder has a degree less than the divisor, the division is likely correct.

Quotient and Remainder

When dividing polynomials, two important parts of the solution are the quotient and the remainder. These can be thought of similarly as the result of whole number division, like when you divide 7 by 3 to get a quotient of 2 and a remainder of 1.

• The quotient is the polynomial that results when you divide the dividend by the divisor fully, ignoring any remainder initially.
• The remainder is what’s left when the polynomial division cannot continue because the degree of the remaining polynomial is less than the degree of the divisor.

In the original exercise, the quotient was found to be \(x^2 + 2x + 1\), and the remainder was \(-4\), with both terms forming the complete solution \(x^2 + 2x + 1 - \frac{4}{x^2 - 4}\). This format: Quotient plus \(\frac{Remainder}{Divisor}\) gives a clear and complete result of polynomial division.

Algebraic Expressions

Algebraic expressions form the foundation of polynomial division and other mathematical operations that involve symbols. Understanding how expressions work can make it easier to handle more complex algebraic manipulations.

• An expression is made up of variables, coefficients (numbers), and operations (like addition, subtraction, etc.).
• In polynomial division, both the dividend and the divisor are algebraic expressions, often with multiple terms of varying degrees.

By mastering the basics of algebraic expressions, students can more easily understand how to manipulate them, as seen during the process of setting up long division for polynomials and solving them step-by-step.