Problem 8 - Elementary and Intermediate Algebra

Question

\((3 a)\left(2 a^{2}\right)\) can be classified as a monomial - monomial. Classify the following products by identifying the types of polynomial factors. a. \(6 x(x-7)\) b. \((9 a+1)(5 a-3)\) c. \((c-d)\left(c^{2}-c+d\right)\) d. \(6 m\left(m^{2}+1\right)\left(m^{2}-1\right)\)

Step-by-Step Solution

Answer

a. Monomial - Binomial, b. Binomial - Binomial, c. Binomial - Trinomial, d. Monomial - Binomial - Binomial.

Step 1: Understanding Polynomials

Polynomials are algebraic expressions consisting of terms. A monomial is a polynomial with one term, a binomial has two terms, and a trinomial has three terms. Identifying how many terms each factor has will help us classify the product of these factors.

Step 2: Classifying Problem a

The expression is \(6x(x-7)\). The first factor \(6x\) is a monomial because it only has one term. The second factor \(x-7\) is a binomial because it has two terms. Therefore, \(6x(x-7)\) is classified as a monomial - binomial.

Step 3: Classifying Problem b

The expression is \((9a+1)(5a-3)\). Both factors \(9a+1\) and \(5a-3\) are binomials because each has two terms. Therefore, \((9a+1)(5a-3)\) is classified as a binomial - binomial.

Step 4: Classifying Problem c

The expression is \((c-d)(c^2-c+d)\). The first factor \(c-d\) is a binomial, having two terms. The second factor \(c^2-c+d\) is a trinomial with three terms. Therefore, \((c-d)(c^2-c+d)\) is classified as a binomial - trinomial.

Step 5: Classifying Problem d

The expression is \(6m(m^2+1)(m^2-1)\). The factor \(6m\) is a monomial, \(m^2+1\) is a binomial, and \(m^2-1\) is also a binomial. We consider the entire expression's classification based on pairwise factors: the combination is first a monomial - binomial, followed by the result as a binomial - binomial. Thus, it begins as a monomial - binomial - binomial.

Key Concepts

Monomial

A monomial is the simplest type of polynomial. It consists of only one term. This single term is a product of a number called the coefficient and a variable raised to a non-negative integer power. Monomials can look very different but have one key thing in common: one term. For example, in the expression \(6x\), "6" is the coefficient and "x" is the variable.

• Examples: \(3a, -5xy, 7x^2y\)
• Structure: Coefficient × Variable(s)

To identify a monomial, count the number of terms. If there's only one, congrats, it’s a monomial! In the classification exercise, for example, the expression \(6x\) is a monomial because it contains only one unit or cluster of numbers and variables stuck together.

Binomial

A binomial is a polynomial made up of two distinct terms. These terms are often separated by a plus (+) or minus (-) sign. A common characteristic of binomials is that they express a certain level of complexity with two separate pieces, yet remain relatively simple. Examples might include expressions like \(x + 3\) or \(2a - 7b\).

• Examples: \(x - 4, 2y + 3\)
• Structure: Term1 ± Term2

In the exercise, an example is \(x - 7\), which is a binomial because it shows two terms. In our classifications, recognizing binomials is crucial as it helps in determining how entire expressions can be grouped or simplified via multiplication or factoring.

Trinomial

A trinomial is a polynomial that includes three terms. This type of polynomial is seen frequently in algebra and can be identified by its three distinct segments. Each term will generally have a different level of complexity or a unique relation to the others. Examples could be simple, like \(a^2 + b + c\), or more complex, such as \(x^2 - 2x + 1\).

• Examples: \(a^2 - 3ab + 4b\), \(x^2 + x + 1\)
• Structure: Term1 ± Term2 ± Term3

In the example from the exercise, \(c^2 - c + d\) is a trinomial because it consists of exactly three terms. Understanding this concept aids in breaking down more complicated expressions into manageable parts, allowing for easier algebraic manipulation and solving.