Problem 23 - Algebra for College Students

Question

Subtract the polynomials using the horizontal format. \(-4 a-5\) from \(6 a+2\)

Step-by-Step Solution

Answer

The result is \( 10a + 7 \).

Step 1: Identify the Polynomials

We need to subtract the polynomial \( -4a - 5 \) from \( 6a + 2 \). This means our expression is \( (6a + 2) - (-4a - 5) \).

Step 2: Remove Parentheses

Distribute the negative sign across the second polynomial. This changes all signs in \( -4a - 5 \): \( 6a + 2 + 4a + 5 \).

Step 3: Combine Like Terms: Coefficients of 'a'

First, focus on the terms with 'a': \( 6a + 4a = 10a \).

Step 4: Combine Like Terms: Constant Terms

Now consider the constant terms: \( 2 + 5 = 7 \).

Step 5: Final Result

Combine the results from like terms: \( 10a + 7 \). Hence, the result of the subtraction is \( 10a + 7 \).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents, combined using addition, subtraction, or multiplication. A polynomial can have multiple terms, such as "6a + 2" or "-4a - 5". In these terms, "6a" and "-4a" are the variable terms, while "2" and "-5" are constant terms. The variable terms have coefficients, which are the numbers in front of the variables, in this case, 6 and -4. Understanding the structure of polynomials helps in manipulating and performing operations on them, such as addition and subtraction. In our exercise, we focus on subtracting one polynomial from another by understanding their terms and how to combine them successfully.

Like Terms

Like terms are terms within an algebraic expression that have the same variables raised to the same powers. For example, in our exercise, the terms "6a" and "-4a" contain the same variable "a" and are both to the first power, making them like terms. This similarity allows us to combine them during arithmetic operations by simply adding or subtracting their coefficients. Identifying and combining like terms is a crucial step in simplifying expressions and solving equations. In our example, once the polynomials are put into a single expression, we find that the like terms "6a" and "4a" can be combined to form "10a". Similarly, the constant terms "2" and "5" are also like terms and can be combined to yield "7".

Horizontal Format

Subtracting polynomials in a horizontal format involves writing the entire expression on a single line. This format shows the subtraction clearly, allowing us to see exactly which terms from one polynomial are being deducted from another. In our specific problem, we express the subtraction of the polynomial \(-4a - 5\) from \(6a + 2\) as \((6a + 2) - (-4a - 5)\). Keeping the expression in horizontal format helps us apply operations systematically and ensures that all terms are accounted for. After distributing any necessary signs, we can then focus on combining like terms without rearranging the expression. This format can be particularly helpful in maintaining clarity when handling complex expressions or attempting to follow each step in a structured manner.

Distribute Negative Sign

The negative sign in front of a polynomial indicates that each term within the polynomial should be changed to its opposite sign. This is known as distributing the negative sign, and it's an important step when subtracting polynomials.In our exercise, the expression \((6a + 2) - (-4a - 5)\) requires us to change \(-4a\) to \(4a\) and \(-5\) to \(5\). The negative sign changes both terms, which then allows us to proceed with combining all terms into a simplified expression. Distributing the negative sign correctly ensures that all terms are correctly positioned for addition, rather than affecting the final results through sign errors. It's a fundamental skill necessary for accurate arithmetic operations in algebra.