Problem 1 - Intermediate Algebra

Question

Fill in the blanks. The expression \(\frac{18 x^{7}}{9 x^{4}}\) is a monomial divided by a _________. The expression \(\frac{6 x^{3}-4 x^{2}+8 x-2}{2 x^{4}}\) is a _________ divided by a monomial. The expression \(\frac{x^{2}-8 x+12}{x-6}\) is a trinomial divided by a _________.

Step-by-Step Solution

Answer

Monomial, Polynomial, Binomial.

Step 1: Simplify into Recognizable Terms

Let's look at the given expressions and identify the forms of numerator and denominator. For the expression \(\frac{18 x^{7}}{9 x^{4}}\):- **Numerator**: \(18 x^7\) is a monomial (a single term consisting of numbers and variables).- **Denominator**: \(9 x^4\) is another monomial. For the expression \(\frac{6 x^{3}-4 x^{2}+8 x-2}{2 x^{4}}\):- **Numerator**: \(6x^3 - 4x^2 + 8x - 2\) is a polynomial with four terms (quartic polynomial).- **Denominator**: \(2x^4\) is a monomial.For the expression \(\frac{x^{2}-8 x+12}{x-6}\):- **Numerator**: \(x^2 - 8x + 12\) is a trinomial (a polynomial with three terms).- **Denominator**: \(x - 6\) is a binomial (a polynomial with two terms).

Step 2: Fill in the Blanks Based on Identified Terms

Now that we have identified the terms, we can proceed to fill in the blanks. 1. For \(\frac{18 x^{7}}{9 x^{4}}\), a monomial is divided by a **monomial**.2. For \(\frac{6 x^{3}-4 x^{2}+8 x-2}{2 x^{4}}\), a polynomial (or quartic polynomial) is divided by a **monomial**.3. For \(\frac{x^{2}-8 x+12}{x-6}\), a trinomial is divided by a **binomial**.

Key Concepts

Monomials

A monomial is one of the most straightforward forms of polynomials. It consists of a single term, which is a product of numbers and variables. This simplicity makes monomials a foundational concept in algebra.
A monomial has no addition or subtraction within it, only multiplication. For example, the expression \(18x^7\) is a monomial because it contains the coefficient 18 and the variable base \(x\) raised to the power of 7. Similarly, \(9x^4\) is another example of a monomial.

• A monomial has only one term.
• It might have multiple variables, each raised to a power.
• When two monomials are divided, as in \(\frac{18x^7}{9x^4}\), the result is another monomial after simplifying by reducing coefficients and subtracting exponents.

Binomials

Binomials are a more complex type of polynomial, consisting of exactly two terms. These terms are separated by an addition or subtraction sign. Binomials are common in algebra and have many practical applications and uses.
An example of a binomial is \(x-6\), which includes two terms: \(x\) and \(6\). Each term is independent and contributes to the binomial's structure.

• Binomials consist of two distinct terms.
• They can involve variables, constants, and both can have their exponents.
• Operations with binomials often appear in factorization problems and division, like in dividing a trinomial \(x^2 - 8x + 12\) by the binomial \(x-6\).

Trinomials

Trinomials are polynomials made up of three terms. These terms are also separated by addition or subtraction signs. Trinomials frequently appear in algebra and are integral to understanding more complex polynomial expressions.
For instance, the trinomial \(x^2 - 8x + 12\) includes three terms: \(x^2\), \(-8x\), and \(12\). Each component of a trinomial has its own distinct role in the expression's value.

• Trinomials contain exactly three terms.
• These terms can vary in degree, typically having a descending order of exponents.
• Dividing trinomials, such as the division of \(x^2 - 8x + 12\) by \(x-6\), involves understanding how each term interacts with the divisor.