Problem 70 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a trinomial in \(x\) of degree 6 is divided by a trinomial in \(x\) of degree \(3,\) the degree of the quotient is 2

Step-by-Step Solution

Answer

The statement is false. When a trinomial of degree 6 is divided by a trinomial of degree 3, the degree of the quotient is 3, not 2. To make the statement true, it should be: If a trinomial of degree 6 is divided by a trinomial of degree 3, the degree of the quotient is 3.

Step 1: Understanding the Problem

The given statement is that if a trinomial of degree 6 is divided by a trinomial of degree 3, the degree of the quotient is 2. The degree of a polynomial is the maximum degree of its terms.

Step 2: Applying Polynomial Division Rule

The degree of the quotient in a polynomial division is the difference between the degrees of the dividend and the divisor. Here, the dividend is the trinomial of degree 6, and the divisor is the trinomial of degree 3.

Step 3: Determining the Degree of the Quotient

Therefore, the degree of the quotient would be \(6 - 3 = 3\), not 2 as stated.

Step 4: Correction of the Statement

To make the statement true, the degree of the quotient should be stated as 3 instead of 2.

Key Concepts

Degree of a Polynomial

When we talk about the degree of a polynomial, we are referring to the highest power of the variable in its expression. In simpler terms, it is the largest sum of exponents that appear in any one term of the polynomial.
For example, in the polynomial \(2x^6 + 4x^3 + 5x\), the degree is 6, because \(x^6\) is the term with the highest exponent.

• The degree helps us understand the polynomial's behavior and its graph.
• It's a valuable tool for predicting the roots and shape of the curve.
• A polynomial with a higher degree will usually oscillate more dramatically.

Understanding the degree is essential, especially when dividing polynomials. When you divide polynomials, the degree of the resulting quotient depends on the degrees of both the dividend and the divisor.

Trinomial

A trinomial is a special type of polynomial that consists of exactly three terms. These terms are typically added or subtracted from one another.
For example, \(x^6 + 3x^3 + 2\) is a trinomial because it has three distinct terms.

• Trinomials are often used in polynomial division problems.
• They are simple to work with and provide a clear insight into the basic structure of polynomial expressions.
• In a trinomial, each term can have a different coefficient and degree.

When you are dividing a trinomial by another trinomial, it's crucial to pay attention to the degrees of each to determine the degree of the quotient.

Quotient

The quotient is the result you get when you divide one polynomial by another. It’s similar to division with numbers, where the dividend is divided by the divisor to give a quotient.

• During polynomial division, the remainder may or may not be zero.
• The quotient, along with the remainder, tells us how the dividend relates to the divisor.
• Understanding the quotient is key to mastering polynomial division.

The degree of this quotient depends on the degrees of the polynomials involved. When dividing a polynomial by another, the degree of the quotient is determined by subtracting the degree of the divisor from the degree of the dividend.

Degree of the Quotient

The degree of the quotient is an important concept in polynomial division. It provides a straightforward way to understand how the degrees of the dividend and divisor affect the outcome of the division.
In the example given in the exercise, a trinomial of degree 6 was divided by a trinomial of degree 3. To find the degree of the quotient, use the rule: \[ \text{Degree of Dividend} - \text{Degree of Divisor} = \text{Degree of Quotient} \] Here, this calculation is \(6 - 3 = 3\).

• This tells us that the degree of the quotient is 3, not 2 as stated in the beginning.
• This formula is vital for correcting misconceptions in polynomial division.

It allows for quick and easy correction of the degree of the quotient if initially guessed incorrectly, ensuring more accurate solutions in polynomial-related problems.