Problem 3 - Elementary and Intermediate Algebra

Question

The expression \(\frac{x^{2}-8 x+12}{x-6}\) is a trinomial divided by _________.

Step-by-Step Solution

Answer

The expression is a trinomial divided by a binomial.

Step 1: Identify the Structure of the Expression

The given expression is \( \frac{x^{2} - 8x + 12}{x - 6} \). It consists of a polynomial in the numerator, particularly a quadratic trinomial, and a linear polynomial in the denominator.

Step 2: Identify the Type of Polynomial

In the numerator, \( x^2 - 8x + 12 \) is a trinomial, which means it is a polynomial with three terms. The denominator \( x-6 \) is a binomial since it has two terms.

Step 3: Answer the Question

The question asks about the polynomial type in the denominator. The denominator \( x-6 \) is a binomial as it consists of two terms.

Key Concepts

Trinomial

Understanding a trinomial will help you grasp the larger concept of polynomial expressions. A trinomial is a type of polynomial that contains exactly three terms. For instance, in the expression \( x^2 - 8x + 12 \), each term is separated by a plus or minus sign.

• The first term is \( x^2 \), which is a quadratic term because it has a degree of 2.
• The second term is \(-8x \), known as the linear term because its degree is 1.
• The third term is \(12 \), referred to as the constant term for having a degree of 0.

Together, these three terms form a trinomial with each term influencing the shape of the graph and the equation's solution. A good approach is always to identify each term's role within the trinomial.

Binomial

A binomial is a polynomial consisting of two distinct terms. Often, binomials are used in algebraic expressions for operations like division and multiplication.
In our exercise, the denominator \( x-6 \) is a binomial. This is because it has:

• The first term: \( x \) which carries a power of 1, making it linear.
• The second term: \(-6 \), a constant term.

Understanding binomials is crucial because they often act as factors in polynomial division. By recognizing the two parts of a binomial, you can solve and simplify expressions more effectively.

Polynomial Division

Polynomial division helps simplify expressions where a polynomial in the numerator is divided by another polynomial in the denominator. In the given exercise, we divide a trinomial, \( x^2 - 8x + 12 \), by a binomial, \( x-6 \).
The process resembles long division with numbers:

• Determine how many times the leading term of the denominator goes into the leading term of the numerator.
• Multiply this result by the entire denominator and subtract from the original polynomial.
• Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the denominator.

This technique not only simplifies the expression but also reveals factors and roots of the polynomial, offering deeper insights into its structure and behavior.