Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Divide. $$\frac{3 x^{4}-7 x^{3}+6 x-16}{3 x-7}$$

Step-by-Step Solution

Answer

The quotient is \(x^3 + 2\) with a remainder of \(-2\).

Step 1: Set Up Polynomial Long Division

To divide using polynomial division, set up your division similar to regular long division. Place the dividend \(3x^4 - 7x^3 + 0x^2 + 6x - 16\) inside the division symbol and the divisor \(3x - 7\) outside.

Step 2: Divide the Leading Terms

Divide the first term of the dividend \(3x^4\) by the first term of the divisor \(3x\) to get \(x^3\). This will be the first term of our quotient.

Step 3: Multiply and Subtract

Multiply \(x^3\) by the entire divisor \(3x - 7\), resulting in \(3x^4 - 7x^3\). Subtract this from the original dividend, which results in \(0x^4 + 0x^3 + 0x^2 + 6x - 16\).

Step 4: Bring Down the Next Term

Since the subtraction results in all zero coefficients for the first two terms, bring down the next term from the dividend, which is \(6x\), resulting in \(6x - 16\).

Step 5: Repeat Division Process

Divide the first term \(6x\) by the leading term of the divisor \(3x\) to obtain \(2\). Multiply \(2\) by the entire divisor \(3x - 7\) to get \(6x - 14\). Subtract \(6x - 14\) from \(6x - 16\) to get \(-2\).

Step 6: State the Result

The result of the division is the quotient \(x^3 + 2\) with a remainder of \(-2\). Hence, \(\frac{3x^4 - 7x^3 + 6x - 16}{3x - 7} = x^3 + 2 - \frac{2}{3x - 7}\).

Key Concepts

Quotient and Remainder

When we perform polynomial long division, just like in regular division, we aim to find both the quotient and the remainder. The quotient is the result we obtain that can be multiplied by the divisor to get close to the original polynomial. The remainder is what is left over after all the division steps are complete.

To illustrate, in our polynomial division exercise where we have divided \( \frac{3x^4 - 7x^3 + 6x - 16}{3x - 7} \), we ended up with a quotient of \( x^3 + 2 \) and a remainder of \(-2\).

Understanding both the quotient and the remainder helps us to express the result of the division as:

• Quotient: The main result of the division process.
• Remainder: What is left over that cannot be divided further by the divisor.

In our division, the complete expression after division is written as:\[ x^3 + 2 - \frac{2}{3x - 7} \]The last term represents the unprocessable part of the division, where \(-2\) is our remainder.

Step-by-Step Solutions

Breaking down problems into step-by-step solutions is key to understanding polynomial long division. This approach simplifies seemingly complex tasks into manageable actions, making them more approachable for learners.

Here's a concise review of how each step transforms the division process:

• **Step 1**: Set up the division, placing the dividend under the bracket and the divisor outside.
• **Step 2**: Focus on dividing the leading terms first. This gives us the first term of the quotient.
• **Step 3**: Multiply the current quotient term with the divisor, subtract the result from the current dividend.
• **Step 4**: Bring down the next term, simplifying if needed, and repeat the process.
• **Step 5**: Continue until every term in the original dividend has been used.
• **Step 6**: Finally, state the result with both quotient and remainder.

By systematically following these steps, you simplify the division task, making complex problems much clearer and enabling better learning outcomes.

Polynomial Division Process

Polynomial division process is a systematic method like long division for numbers, adapted for dividing polynomials. This process helps break down higher degree polynomials into simpler components, enabling easier manipulation and solution of equations.

To execute polynomial division successfully:

• Ensure both dividend and divisor are in descending order of their power.
• Keep track of all terms, including those with coefficients of zero, such as \(0x^2\) in our example.
• Each step involves dividing, multiplying, subtracting, and then repeating.

Often, it helps to write out each polynomial term clearly, so mistakes are avoided when aligning terms during division and subtraction.

The ultimate aim is to reduce the polynomial to as simple a quotient and remainder as possible. Understanding each step enhances confidence in handling polynomials and extends to solving more complex algebraic expressions.