Problem 58 - Beginning and Intermediate Algebra

Question

Subtract. $$\begin{array}{r} -3 d^{2}+16 d+2 \\ -\quad 5 d^{2}+7 d-3 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The short answer to the subtraction of the given expressions is: \(-8d^2 + 23d + 5\).

Step 1: Write down the expressions

First, write down the given expressions: 1st Expression: \(-3d^2 + 16d + 2\) 2nd Expression: \(5d^2 - 7d -3\)

Step 2: Identify the coefficients of each term

We will identify the coefficients of the terms: 1st Expression: \(d^2\) term: \(-3\) \(d\) term: \(16\) Constant term: \(2\) 2nd Expression: \(d^2\) term: \(5\) \(d\) term: \(-7\) Constant term: \(-3\)

Step 3: Subtract corresponding coefficients

Now, subtract the coefficients of the two expressions term-by-term: \(d^2\) term: \(-3 - 5 = -8\) \(d\) term: \(16 - (-7) = 16 + 7 = 23\) Constant term: \(2 - (-3) = 2 + 3 = 5\)

Step 4: Write the resulting expression

Combine the coefficients back with their respective terms: \(-8d^2 + 23d + 5\)

Step 5: Solution

The subtraction of the given expressions is: \(-8d^2 + 23d + 5\)

Key Concepts

Coefficient Identification

In polynomials, the coefficient is the number that multiplies the variable(s) in a term. For example, in the term \(-3d^2\), \(-3\) is the coefficient. Identifying coefficients correctly is crucial for operations like polynomial subtraction.

Polynomials often contain multiple terms, and each can have its own distinct coefficient. In our example, we have two expressions. By looking closely at the terms, you can pick out the coefficients:

• \(d^2\) term coefficients are \(-3\) and \(5\).
• \(d\) term coefficients are \(16\) and \(-7\).
• The constant coefficients (terms without a variable) are \(2\) and \(-3\).

Knowing coefficients sets the stage for accurate subtraction, as each must be correctly paired with its counterpart from the other expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators (such as plus or minus). They are like mathematical phrases that can be simplified or manipulated. In our example: \(-3d^2 + 16d + 2\) and \(5d^2 - 7d -3\), these are both algebraic expressions.

Here are a few characteristics of algebraic expressions:

• They often contain one or more terms—each consisting of a coefficient and variable(s).
• The terms are separated by operators like "+" or "-".

Understanding the structure of an algebraic expression helps in tasks like "term-by-term subtraction," where you need to match each type of term to perform accurate operations. This task involves lining up and subtracting matching terms, ensuring the resulting expression is simplified and correct.

Term-by-Term Subtraction

Subtraction of polynomials involves subtracting the corresponding coefficients term by term. Let's explore this method using our example.

First, line up the like terms from the two expressions, which are those with the same variable degree:

• The \(d^2\) terms: \(-3d^2\) and \(5d^2\).
• The \(d\) terms: \(16d\) and \(-7d\).
• The constant terms: \(2\) and \(-3\).

Next, perform subtraction by subtracting the coefficients of these matching terms:

• For \(d^2\) terms: \(-3 - 5 = -8\), resulting in \(-8d^2\).
• For \(d\) terms: \(16 - (-7) = 16 + 7 = 23\), resulting in \(23d\).
• For constant terms: \(2 - (-3) = 2 + 3 = 5\), resulting in \(5\).

The newly formed expression is \(-8d^2 + 23d + 5\). This simplified expression is the result of subtracting the original polynomials on a term-by-term basis. Using this method ensures the correct outcome while working neatly and systematically through each step.