Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Divide. $$\left(-3 x^{3}+2 x^{2}+2 x\right) \div\left(6 x^{2}+2 x+1\right)$$

Step-by-Step Solution

Answer

Quotient: \\(-\frac{1}{2}x + \frac{1}{2}\\), Remainder: \\(\frac{3}{2}x - \frac{1}{2}\\).

Step 1: Setup the Long Division

Write the dividend \(-3x^3 + 2x^2 + 2x\) and the divisor \(6x^2 + 2x + 1\) in the long division format, which resembles the layout you might use for a long division problem involving numbers.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor, \(-3x^3 \/ 6x^2\), which simplifies to \(-\frac{1}{2} x\). Write \(-\frac{1}{2} x\) above the division bar as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(6x^2 + 2x + 1\) by the first term of the quotient \(-\frac{1}{2} x\). The result is \(-3x^3 - x^2 - \frac{1}{2} x\). Subtract this from the original dividend, \(-3x^3 + 2x^2 + 2x\), to get a new polynomial \(3x^2 + \frac{5}{2} x\).

Step 4: Repeat the Process

Now, consider the new polynomial \(3x^2 + \frac{5}{2} x\), and divide its leading term \(3x^2\) by the leading term of the divisor \(6x^2\), which gives \(\frac{1}{2}\). Write \(\frac{1}{2}\) next to \(-\frac{1}{2} x\) in the quotient.

Step 5: Multiply and Subtract Again

Multiply the entire divisor \(6x^2 + 2x + 1\) by \(\frac{1}{2}\) to get \(3x^2 + x + \frac{1}{2}\). Subtract this from the polynomial \(3x^2 + \frac{5}{2} x\), resulting in \(\frac{3}{2}x - \frac{1}{2}\).

Step 6: Conclude the Division

Since the degree of the remaining polynomial \(\frac{3}{2}x - \frac{1}{2}\) is less than the degree of the divisor, it becomes the remainder. Thus, the quotient is \(-\frac{1}{2}x + \frac{1}{2}\) and the remainder is \(\frac{3}{2}x - \frac{1}{2}\).

Key Concepts

Long Division of Polynomials

Long division of polynomials is a systematic method used for dividing a polynomial by another polynomial of equal or lower degree. It extends the idea of simple long division that you might use with numbers but applies it to polynomial expressions. To begin, set up the division in a similar layout to numerical long division:

• Dividend: This is the polynomial you want to divide. In our problem, it's \(-3x^3 + 2x^2 + 2x\).
• Divisor: This is the polynomial you're dividing by, here \(6x^2 + 2x + 1\).

Begin by dividing the leading term of the dividend by the leading term of the divisor. Place the result as the first term of your quotient. Multiply the entire divisor by this term, then subtract the result from the dividend. The process generates a new polynomial, and you repeat the steps until the degree of the residual is less than the degree of the divisor.
This sequence of steps is crucial for getting the correct quotient and remainder, making this method meticulous but highly structured.

Quotient and Remainder

In polynomial division, just like in numerical division, the result comprises two parts: the quotient and the remainder. These two parts are related in the following way:

• Quotient: This is what you obtain when you divide the dividend by the divisor as neatly as possible, excluding anything left over. In our example, the quotient is \(-\frac{1}{2}x + \frac{1}{2}\).
• Remainder: After dividing as much as possible, what is left over is called the remainder. It has a degree that is less than that of the divisor. Here, the remainder is \(\frac{3}{2}x - \frac{1}{2}\).

The remainder is crucial, as it tells us how much doesn't exactly "fit" into our division. The division can be represented by the structure: \[ \text{Dividend} = (\text{Divisor}) \times (\text{Quotient}) + \text{Remainder} \]
Understanding the relation between these parts helps verify whether your division was performed correctly.

Polynomial Simplification

Simplifying polynomials allows us to make equations easier to work with. Simplification can involve factoring, expanding, or reducing expressions to their minimal form, and, in the context of division, ensuring that there's no redundant complexity in either the quotient or remainder.
The simplification occurs throughout the division process when you're continuously dividing, multiplying, and subtracting polynomial terms. For instance, reducing fractions like \(-\frac{3}{6x}\) to \(-\frac{1}{2}x\) simplifies the work.

• Remember to combine like terms as you proceed, which means adding or subtracting coefficients of terms with the same power of x.
• If the remainder has like terms, simplify these to ensure the simplest form possible.
• Always double-check that fractions within coefficients are in their simplest form.

By adhering to simplification principles during long division, you'll make the process not only more manageable but also more clear and logical, enabling clearer insights into polynomial behavior.