Problem 19 - Algebra and Trigonometry

Question

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{x^{3}+6 x+3}{x^{2}-2 x+2} $$

Step-by-Step Solution

Answer

Quotient: \(x + 2\); Remainder: \(8x - 1\).

Step 1: Arrange the Division

To perform the division, arrange the dividend, \(x^3 + 0x^2 + 6x + 3\), and the divisor, \(x^2 - 2x + 2\). Here, we added \(0x^2\) to ensure all powers of \(x\) are accounted for. Begin by dividing the first term of the dividend by the first term of the divisor.

Step 2: Divide the Leading Terms

Divide \(x^3\) by \(x^2\), which gives \(x\). Write \(x\) as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(x^2 - 2x + 2\) by \(x\) to get \(x^3 - 2x^2 + 2x\). Subtract this from the current dividend \(x^3 + 0x^2 + 6x + 3\) to get a new remainder \(2x^2 + 4x + 3\).

Step 4: Repeat the Process for New Terms

Next, divide the leading term of the new remainder, \(2x^2\), by the leading term of the divisor, \(x^2\). The result is \(2\). Add \(2\) to the quotient.

Step 5: Multiply and Subtract Again

Multiply the entire divisor \(x^2 - 2x + 2\) by \(2\) to get \(2x^2 - 4x + 4\). Subtract this from the remainder \(2x^2 + 4x + 3\) to find the new remainder, which is \(8x - 1\).

Step 6: Check for Completion

Since the degree of the new remainder \(8x - 1\) is less than the degree of the divisor \(x^2 - 2x + 2\), the division is complete. The quotient is \(x + 2\) and the remainder is \(8x - 1\).

Key Concepts

Understanding Dividend and Divisor

In polynomial long division, two main components play crucial roles: the **dividend** and the **divisor**. They work together to help you find the quotient and remainder.

• The dividend is the polynomial you are dividing. In our exercise, the dividend is \( x^3 + 6x + 3 \).
• The divisor is the polynomial by which you divide the dividend. Here, it is \( x^2 - 2x + 2 \).

To simplify the division, make sure to include all powers of \( x \) in the dividend, even if that means adding a term with a coefficient of 0, such as \( 0x^2 \) in our example. This accounting keeps the process organized and manageable. By dividing the leading terms of the dividend and divisor, you start forming the quotient.

Finding the Quotient and Remainder

The goal of polynomial long division is to express the dividend as the product of the divisor and the quotient added to the remainder. This method is similar to the long division you use with numbers.

• The quotient is the result of the division. In our case, after dividing step-by-step, we get \( x + 2 \).
• The remainder is what's left over after division. For our problem, the remainder is \( 8x - 1 \).

To find these, continually divide the leading term of the current remainder by the divisor's leading term. Each time, multiply the entire divisor by the result and subtract it from the current remainder. Repeat until the degree of the remainder is less than that of the divisor.

Exploring the Degree of Polynomial

In polynomial division, understanding the **degree of a polynomial** is essential. The degree helps determine when you've completed the division.- The **degree** of a polynomial is the highest power of the variable in the expression. For example, \( x^3 \) has a degree of 3.- The degree affects how you structure the division process. Always start dividing with the highest degree terms.In the given problem:

• The dividend \( x^3 + 6x + 3 \) has a degree of 3 because of \( x^3 \).
• The divisor \( x^2 - 2x + 2 \) has a degree of 2.
• When the remainder has a degree less than the divisor, as in \( 8x - 1 \) (degree 1), the division process is complete.

Knowing these degrees at each step ensures you keep the division process consistent and accurate.