Simple Algebraic Division

Overview

Algebraic division is a process similar to numerical division, where one algebraic expression (the dividend) is divided by another (the divisor).
The result is a quotient, which may also have a remainder.
Essentially, you’re finding how many times the divisor ‘fits’ into the dividend, similar to long division with numbers.

Step 1: Write the dividend and divisor down in descending powers of the variable.
Step 2: Divide the leading term (highest power term) of the dividend by the leading term of the divisor, write this in the quotient.
Step 3: Multiply the divisor by the first term of your quotient and subtract this from the original dividend. Bring down the next term from the dividend.
Step 4: Repeat step 2 and 3 until you cannot divide any further.
What is left at the end is referred as to the remainder.
Note: If the divisor does not divide exactly into the dividend the remainder should be kept as a fraction rather than converted into a decimal.

Algebraic division is a critical method in simplifying expressions involving polynomials. This is particularly important in understanding and manipulating identities and inequalities.
Simplifying algebraic inequalities often involves the division of polynomials. For instance, if a greater degree polynomial is divided by a lesser degree polynomial, this can help in exploring the inequality’s properties and solving it more easily.
It’s also a vital step in polynomial long division, used when dealing with more complex identities and inequalities.

A mistake often made when starting out is not writing the dividend and divisor in descending order of powers. For example, the polynomial 3x^2 + 5x + 2 should be written as is, not 5x + 3x^2 + 2.
Another common mistake is forgetting to subtract the result of the multiplication (in step 3). It’s easy to accidentally add instead of subtract when you’re working through the process.
Last, it’s crucial when subtracting polynomials to remember to distribute the negative sign to all terms of the polynomial before subtracting them. For example, when subtracting (3x^2 + x - 2), subtract each term individually: minus 3x^2, minus x, minus (-2).

Now that you understand the process and common pitfalls of simple algebraic division, it’s time to practice. Drill problems until you feel comfortable with the process. Remember, consistent practice is key to mastering this concept.