Problem 34 - Algebra 2

Question

Simplify. Classify each result by number of terms. $$ \left(-8 d^{3}-7\right)+\left(-d^{3}-6\right) $$

Step-by-Step Solution

Answer

\(-9d^3 - 13\), which is a binomial.

Step 1: Simplify the Polynomial

Combine the like terms by adding the coefficients of the terms with the same variable and exponent. In this case, add the coefficients of the terms that contain the variable to the power of 3, which are \( -8d^3 \) and \( -d^3 \) to get \( -9d^3 \) and then combine the constant terms \( -7 \) and \( -6 \) to get \( -13 \)

Step 2: Write the Simplified Expression

Combine the results of the like terms to form the simplified expression. The simplified polynomial is \( -9d^3 - 13 \)

Step 3: Classify by Number of Terms

The simplified expression \( -9d^3 - 13 \) has two terms, which makes it a binomial.

Key Concepts

Binomial

A binomial is a polynomial which is the sum of exactly two terms, each of which is a monomial, that is, a number, a variable, or a product of numbers and variables. In the given example, we aim to simplify the expression to see if the result is indeed a binomial. By definition, each monomial within a binomial is called a term, and these terms can be numbers, variables, or the product of numbers and variables raised to positive integer powers.

For instance, in the simplified expression (-9d^3 - 13), there are two distinct terms: (-9d^3) and (-13). The term (-9d^3) is a monomial that includes a variable (d) raised to the third power, and (-13) is a monomial represented by a constant. Since there are only two terms after simplification, the expression is classified as a binomial.

Polynomial Simplification

Polynomial simplification involves combining like terms to create a simpler equivalent expression. Like terms are terms that have the exact same variable parts, including the same variables and the same exponents. The process simplifies the original polynomial by adding or subtracting the coefficients of these like terms. It's crucial to identify all the like terms correctly before performing any addition or subtraction.

In the given exercise, the process of simplifying the polynomial begins by identifying and combining the terms with the variable d raised to the power of 3. The polynomial simplification does not alter the degree of the polynomial; it only makes the expression more succinct. Ultimately, the goal is to express the polynomial with as few terms as possible while maintaining equivalence with the original expression.

Combining Like Terms

Combining like terms is a crucial step in polynomial simplification. It aids in reducing complexity and making algebraic expressions easier to work with. When combining like terms, you add or subtract the coefficients and keep the variable part of the term unchanged.

In our exercise, the terms (-8d^3) and (-d^3) are like terms because they both contain the variable d raised to the third power. We combine them by adding their coefficients: (-8) and (-1), resulting in (-9d^3). Similarly, we combine the constants (-7) and (-6), since they are like terms without variables, resulting in (-13).

The act of combining like terms is an elemental skill in algebra, and mastering it is essential for efficiently simplifying polynomials and solving various algebraic equations.