Problem 40 - Elementary and Intermediate Algebra

Question

Factor. $$ 4(x+y)+t(x+y) $$

Step-by-Step Solution

Answer

The expression factors to \((x+y)(4+t)\).

Step 1: Identify Common Factor

First, let's look at the expression \( 4(x+y) + t(x+y) \). Notice that \( (x+y) \) is a common term in both parts of the expression.

Step 2: Extract the Common Factor

Since \( (x+y) \) appears in both terms, we can 'factor it out'. This means we'll rewrite the expression as \( (x+y)(4+t) \). Here, \( 4 \) and \( t \) are the coefficients left after factoring out \( (x+y) \).

Step 3: Verify the Factoring

To verify, distribute \( (x+y) \) back into the expression: \( (x+y)(4+t) = 4(x+y) + t(x+y) \), which matches the original expression.

Key Concepts

Common Factor

A common factor is a term that appears in multiple parts of an expression and can be factored out to simplify the expression. Consider the expression \( 4(x+y) + t(x+y) \). Here, the term \((x+y)\) is repeated in both portions of the expression. By identifying this repeated group as a common factor, you can extract it from the entire expression.

When we factor out the common factor \((x+y)\) out of \( 4(x+y) + t(x+y) \), we are essentially simplifying the expression into two components multiplied together: \((x+y)(4+t)\). This method keeps the essence of the expression intact while making it more concise. Recognizing and using common factors can greatly aid in simplifying algebraic expressions and solving more complex equations. Here are the steps to finding common factors:

• Identify terms that appear in each part of the expression.
• Factor out these terms to rewrite the expression in a simplified form.
• Ensure by reversing the operation that the factoring is correct.

Distributive Property

The distributive property is a fundamental principle that allows us to expand or factor expressions. It states that a number multiplied by a sum can be expressed as the sum of the product of the number and each term of the sum. For example, \( a(b+c) = ab + ac \).

In the context of factoring, the distributive property is applied in reverse. If you look at \( 4(x+y) + t(x+y) \), you can apply the distributive property "backwards" by factoring out the common term \((x+y)\): transforming it into \((x+y)(4+t)\). This reverse application helps simplify expressions and solve equations more effectively. Here’s a quick reminder of how the distributive property works:

• When multiplying, distribute the number to every term inside the parentheses.
• To factor, find a common term in multiplied expressions, then take it outside the parentheses.
• Test your simplified expression by expanding it back to check its accuracy.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication) that represent specific values or relationships. In the expression \( 4(x+y) + t(x+y) \), we see variables \( x \) and \( y \), constants like \( 4 \), and a variable \( t \).

An expression can consist of one or more terms. Each term in an algebraic expression could be a standalone number, variable, or a combination of both multiplied together. While simplifying these expressions may seem daunting, recognizing patterns such as common factors and using properties like distributive and associative properties makes it feasible. Understanding these expressions paves the way for solving algebra equations, performing computations, and analyzing relationships in mathematics.

• Algebraic expressions can be simplified by recognizing common patterns or factors.
• Constituents of expressions include variables, constants, and operators.
• Simplifying expressions helps with clarity and sets the foundation for solving algebraic problems.