Question

Use long division to divide. Divisor \(2 x+3\) Dividend $$8 x-5$$

Short answer

The result of the division is \(4 - \frac{17}{2x+3}\)

Step-by-step solution

Set up the long division

Write the problem in long division format as \[\frac{8x-5}{2x+3}\] . The divisor (the term we are dividing by) is on the outside and the dividend (the term we are dividing into) is on the inside.

Divide the first term in the dividend, by the first term in the divisor

Divide the first term in the dividend, which is \(8x\), by the first term in the divisor which is \(2x\). This results in \(4\). This quotient is placed on top as the first part of the answer.

Multiply the divisor by the first term in the quotient and subtract from dividend

Multiply \(2x+3\) by \(4\), which gives \(8x+12\), then Subtract this from \(8x-5\) to get \(-17\). The division process will stop here because the degree of the divisor (which is 1) is greater than the degree of the reminder.

Write down the final result

The final result of the division is \(4\) and the remainder is \(-17\). So the division can be expressed as \(4 - \frac{17}{2x+3}\)

Key concepts

Dividing Polynomials

Dividing polynomials is a fundamental concept in algebra that is very similar to the long division you might do with numbers. When you divide one polynomial by another, the objective is to find how many times the divisor polynomial fits into the dividend polynomial, as well as to determine the remainder, if there is one.

Here's a helpful analogy: think of dividing polynomials like you divide a cake among friends. The dividend is the cake, the divisor is the number of friends, and the quotient is how much each friend gets. Any leftover, of course, would be the remainder - a smaller piece that couldn't be evenly distributed.

• The dividend is the polynomial that you want to divide.
• The divisor is the polynomial that you are dividing by.
• The quotient is the resulting polynomial after the division.
• The remainder is what's left after the division.

In the given exercise, we are dividing the polynomial \(8x - 5\) by \(2x + 3\). It is important that the polynomial terms are organized by decreasing power of the variables before we begin dividing. This organization ensures each step of the division process is executed smoothly.

Algebraic Division

Algebraic division using long division involves several systematic steps. It might seem challenging at first, but it's much like traditional long division with numbers. The process is important for simplifying complex algebraic expressions and polynomials.

Here’s how you should approach algebraic division:

• Begin by identifying the highest degree terms in both the dividend and the divisor.
• Divide the leading term of the dividend by the leading term of the divisor.
• This result is part of your quotient.
• Multiply back and subtract from your original dividend, just like you would in numeric division.
• Continue this process with the new resulting polynomial until you reach a remainder that cannot be divided further.

Following these steps helps ensure accuracy in language expressions and enables you to manage the structure of your algebraic equations.
In the exercise given, dividing the first terms, \(8x\) by \(2x\), gives the first part of the quotient, which is \(4\). The steps require performing operations like multiplication and subtraction repeatedly until the division is complete.

Remainder in Algebraic Division

The remainder in algebraic division gives us information about what is left over after the division process is complete. In many cases, this remainder is a polynomial of a lower degree than the divisor.

Understanding the remainder can be crucial, especially when you're trying to verify your work or when applying concepts such as the Remainder Theorem in more advanced algebra.

• After completing the long division, the remainder is what remains.
• In polynomial long division, if the remainder is 0, it means the divisor can completely divide the dividend evenly.
• When the remainder is not 0, like in this exercise, it needs to be expressed as a fraction of the divisor.

In the provided example, once the process stops, the final remainder after subtracting the terms is \(-17\). Since the degree of the divisor \(2x + 3\) is greater than that of the remainder, the division halts here. The expression of the result includes this remainder, represented as \(-\frac{17}{2x+3}\), which captures the leftover part of the division.