Problem 73 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \left(6 a^{4}-2 a^{3}-3 a^{2}+a+4\right) \div(3 a-1) $$

Step-by-Step Solution

Answer

Answer: The result is \(2a^3 - \frac{1}{3}a^2 + \frac{(\frac{2}{3}a + 4)}{3a - 1}\).

Step 1: Set up the long division

Write the dividend, \((6a^4 - 2a^3 - 3a^2 + a + 4)\), inside the division symbol, and write the divisor, \((3a - 1)\), outside the division symbol, preparing for a long division process.

Step 2: Divide the first term of dividend by the first term of divisor

For the first term of the dividend, which is \(6a^4\), divide it by the first term of the divisor, which is \(3a\). The quotient is \(2a^3\).

Step 3: Multiply the divisor by the quotient

Multiply \((3a-1)\) by the achieved quotient term, \(2a^3\). We get \(6a^4 - 2a^3\).

Step 4: Subtract the result from the current dividend

Line up the obtained result \(6a^4 - 2a^3\) under the dividend \((6a^4 - 2a^3 - 3a^2 + a + 4)\) and subtract. We get \((-1a^2 + a + 4)\) as the remainder.

Step 5: Divide the highest degree term of the new remainder by the first term of divisor

Divide the term with the highest degree in the remainder, \(-a^2\), by the first term in the divisor, \(3a\). The quotient is \(-\frac{1}{3}a^2\).

Step 6: Multiply the divisor by the quotient

Multiply \((3a-1)\) by the achieved quotient, \(-\frac{1}{3}a^2\). We get \(-a^2 + \frac{1}{3}a\).

Step 7: Subtract the result from the current remainder

Line up the obtained result \((-1a^2 + \frac{1}{3}a)\) under the current remainder \((-1a^2 + a + 4)\), and subtract. We get the new remainder as \((\frac{2}{3}a + 4)\). Since the degree of the new remainder is strictly less than the degree of the divisor, we cannot proceed with the division process any further.

Step 8: Write the final answer

The final answer is the quotient terms plus the remainder of the last step divided by the divisor. Thus, the final answer is: $$2a^3 - \frac{1}{3}a^2 + \frac{(\frac{2}{3}a + 4)}{3a - 1}$$

Key Concepts

Long Division in Polynomials

Long division in polynomials is similar to the procedure used for dividing numbers, although it involves expressions with variables. Here, the goal is to find how many times a polynomial (the divisor) fits into another polynomial (the dividend). This process helps understand the division of expressions and is an essential tool in algebra.

• Just like with numbers, align terms by degree from highest to lowest when setting up the division.
• Follow the same repetitive steps: divide, multiply, and subtract until you can't go further.

This division method offers a structured way to break down complex polynomials into simpler components, ensuring clarity and accuracy in results.

Understanding Dividends and Divisors

In polynomial long division, the terms hold specific roles: the dividend and the divisor. The dividend is the polynomial you are dividing, while the divisor is the polynomial you are dividing by.

In our example,

• The dividend is the polynomial \(6a^4 - 2a^3 - 3a^2 + a + 4\).
• The divisor is \(3a - 1\).

To ensure precision in alignment, always arrange these polynomials in descending order of their degree, which refers to the highest power of the variable. By doing this, the division process becomes more manageable and systematic.

The Quotient: Results of the Division

The quotient in polynomial long division is the polynomial that represents how many times the divisor can be multiplied to fit into the dividend. In other words, it's the answer you get when you divide one polynomial by another.

Throughout the division process:

• You find each term of the quotient by dividing the first term of the remaining polynomial by the first term of the divisor.
• The quotient's degree will generally be the difference between the degrees of the dividend and the divisor.

In our solution, the quotient turns out to be \(2a^3 - \frac{1}{3}a^2\). This quotient gives a partial answer to the division before considering the remainder.

Interpreting the Remainder

In division, including polynomial division, the remainder is what's left after the divisor has been multiplied by the quotient and subtracted from the dividend. It should have a lesser degree than the divisor, meaning further division isn't feasible.

In the given exercise:

• The remainder we ended up with is \(\frac{2}{3}a + 4\).
• It's expressed as part of the final answer: \(\frac{(\frac{2}{3}a + 4)}{3a - 1}\).

Understanding remainders is crucial, as they represent what's left in division and are often included in the final expression. This inclusion ensures the entire original polynomial is accounted for in the division process.