Problem 1 - Beginning and Intermediate Algebra

Question

Label the dividend, divisor, and quotient of each division problem. $$\frac{12 c^{3}+20 c^{2}-4 c}{4 c}=3 c^{2}+5 c-1$$

Step-by-Step Solution

Answer

The dividend is \(12 c^{3}+20 c^{2}-4 c\), the divisor is \(4 c\), and the quotient is \(3 c^{2}+5 c-1\).

Step 1: Identify the dividend, divisor, and quotient

In a division problem, the dividend is the number (or expression) being divided, the divisor is the number (or expression) you are dividing by, and the quotient is the result you get at the end of the process. In a fraction, the dividend is the numerator, the divisor is the denominator, and the quotient is the result of dividing the numerator by the denominator. In our given problem: \[\frac{12 c^{3}+20 c^{2}-4 c}{4 c}\]

Step 2: Label and write the dividend, divisor, and quotient

From the previous step, we can now label the dividend, divisor, and quotient as follows: - Dividend: \(12 c^{3}+20 c^{2}-4 c\) - Divisor: \(4 c\) - Quotient: \(3 c^{2}+5 c-1\) In conclusion, the dividend in our problem is \(12 c^{3}+20 c^{2}-4 c\), the divisor is \(4 c\), and the quotient is \(3 c^{2}+5 c-1\).

Key Concepts

Understanding the Dividend in Polynomial Division

A dividend is a crucial concept in any division problem, and particularly in polynomial division. It is the expression that you are going to divide into smaller parts or segments. To visualize this, think of the dividend as the numerically or symbolically larger expression that will be broken down.
In our example, the dividend is the polynomial expression \(12c^3 + 20c^2 - 4c\). This means that it holds more terms and typically has the highest degree when you compare it to the divisor.

• Highest degree: The degree of a polynomial is the highest exponent of its variable. In this dividend, that degree is 3, as seen in the term \(12c^3\).
• Terms: The dividend contains three terms: \(12c^3\), \(20c^2\), and \(-4c\).

Understanding the structure of the dividend helps break it down and apply division effectively.

Defining the Divisor in Polynomial Division

The divisor is the expression by which we divide the dividend, hence making it essential in understanding polynomial division.In our example, the divisor is simply \(4c\).
This portion of a division problem usually has fewer terms or factors than the dividend and plays a key role in simplifying or breaking down the dividend.

• Simplification: The divisor simplifies the dividend into the quotient.
• Comparison: You often find the divisor having lower degree terms compared to the dividend, as with the example \(4c\).

Remember, for division to provide a meaningful result (a quotient), the divisor should be non-zero. For polynomials, ensure you address any shared common factors to simplify the process before proceeding.

Discovering the Quotient in Polynomial Division

The quotient represents the result of a division problem after dividing the dividend by the divisor.In polynomial terms, the quotient shows how many times the divisor fits into the dividend.
In our worked example, dividing \(12c^3 + 20c^2 - 4c\) by \(4c\) yields the quotient \(3c^2 + 5c - 1\).

• Verification: You can verify your division by multiplying the quotient by the divisor and ensuring the product matches the dividend.
• Simplified form: The quotient is usually in its simplest form indicating completion of the division task.

Understanding the quotient helps you determine if your division was accurate and allows you to conclude the division operation effectively. This step ensures that your polynomial equation is neatly simplified.