Problem 21 - Algebra for College Students

Question

Subtract the polynomials using the horizontal format. \(5 x-2\) from \(3 x+4\)

Step-by-Step Solution

Answer

The result of the subtraction is \(-2x + 6\).

Step 1: Write Down the Polynomials

First, identify the polynomials involved in the subtraction. We are subtracting the polynomial \(5x - 2\) from \(3x + 4\).

Step 2: Set Up the Subtraction

Write the expression as \((3x + 4) - (5x - 2)\) for the subtraction operation. Be sure to place the polynomial that will be subtracted second, in parentheses to remember to distribute the negative sign.

Step 3: Distribute Negative Sign

Distribute the negative sign across the second polynomial: \((3x + 4) - 5x + 2\). Each term inside the parentheses changes its sign.

Step 4: Combine Like Terms

Combine the like terms. Start with the \(x\) terms: \(3x - 5x = -2x\). Then combine the constant terms: \(4 + 2 = 6\).

Step 5: Write the Final Result

Put the result of combining like terms together: \(-2x + 6\). This is the result of subtracting \(5x - 2\) from \(3x + 4\).

Key Concepts

Polynomial Operations

Polynomial operations refer to addition, subtraction, multiplication, and division of polynomials. These operations are fundamental in algebra and help us work with expressions that include variables raised to various powers.
When performing polynomial operations, it is crucial to remember the rules of arithmetic while carefully organizing the terms.

• **Addition and subtraction** often involve organizing like terms to simplify the expression.
• **Multiplication** requires using the distributive property to expand the expression.
• **Division** can be done using long division or synthetic division techniques.

In the context of this exercise, we are focusing on subtraction, where understanding how to manage like terms and distribute negative signs is essential.

Like Terms

Like terms are terms within an algebraic expression that have the same variables raised to the same powers.
These terms can be combined through addition or subtraction to simplify expressions. For example, in the polynomial expression \(3x + 4 - 5x + 2\), \(3x\) and \(-5x\) are like terms because both contain the variable \(x\).

• Like terms have identical variable parts, but their coefficients can differ. In our example, \(3x\) and \(-5x\) have different coefficients.
• The number part (or coefficient) of like terms is what you add or subtract.
• Constant terms, like \(4\) and \(2\), are considered like terms because they are both numbers.

Using like terms, we can streamline expressions by merely combining their coefficients, making them simpler to work with.

Distributive Property

The distributive property is a key algebraic principle that permits the multiplication of two terms by distributing one of them across others inside parentheses.
This property is expressed as \(a(b + c) = ab + ac\). In polynomial subtraction, it helps in correctly applying negative signs when removing parentheses.
For instance, consider the polynomial subtraction in the exercise where we set it up as \((3x + 4) - (5x - 2)\):

• Distribute the negative sign across the second polynomial.
• This results in \((3x + 4) - 5x + 2\). Notice how each term in \(5x - 2\) changes sign due to the negative sign.

Mastering the distributive property is necessary for more complex algebraic manipulations and ensuring calculations are handled with precision.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations such as addition, subtraction, multiplication, and division.
They do not have an equality sign, which differentiates them from equations.

• An **example of an algebraic expression** is \(3x + 4\), which mixes numbers (constants) and variables.
• Polynomials are a specific type of algebraic expression characterized by their structure, consisting of terms with variables raised to whole number exponents.
• Understanding how to manipulate these expressions through operations like addition, subtraction, and distribution is foundational for solving problems throughout algebra.

Grasping the basics of algebraic expressions ensures a solid understanding of how these parts work together to form equations and more sophisticated concepts in mathematics.