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Chapter 3: Problem 45

For Problems \(25-50\), factor completely. $$ x(y+2)+3(y+2) $$

Short Answer

Expert verified
The expression factors to \((y+2)(x+3)\).

Step by step solution

01

Identify Common Factors

Look at the given expression: \(x(y+2) + 3(y+2)\). Notice that the term \((y+2)\) appears in both parts of the expression. This indicates that \((y+2)\) is a common factor of both terms.
02

Factor Out the Common Term

Since \((y+2)\) is the common factor, we can factor it out of the expression. Rewrite the expression as: \((y+2)(x+3)\). This is achieved by taking \((y+2)\) out and writing the expression as a product of \((y+2)\) and the remaining terms which are \(x\) and \(3\).
03

Verify the Factorization

To ensure the factorization is correct, distribute \((y+2)\) back into the terms \((x+3)\). Doing this, we get \(x(y+2) + 3(y+2)\), which is the original expression. This confirms that the factorization \((y+2)(x+3)\) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factors
In mathematics, a common factor is a number or algebraic term that divides two or more numbers or expressions without leaving a remainder. When factoring expressions in algebra, identifying common factors is a crucial first step.

To find a common factor, look at each term in the expression. In our example expression, \(x(y+2) + 3(y+2)\), the term \((y+2)\) appears in both parts. This means that \((y+2)\) is a common factor.

By recognizing common factors, we can simplify expressions, making them easier to work with. This step is like finding common ground between terms to simplify and solve algebraic expressions efficiently.
  • Common factors simplify complex expressions.
  • They help in the process of factoring completely.
  • Spotting them requires careful observation of repeating elements.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. They can represent real-world situations and are used widely in mathematics to solve problems.

In our given problem, \(x(y+2) + 3(y+2)\) is an algebraic expression. This expression has variables (\(x\) and \(y\)), constants (\(3\)), and the arithmetic operations of multiplication and addition.

Understanding algebraic expressions is essential for factoring because it allows us to manipulate the terms to find common factors and simpler expressions. They are:
  • Composed of terms such as constants, coefficients, and variables.
  • Form the building blocks of algebra.
  • Can be rearranged to simplify or solve mathematical problems.
Distributive Property
The distributive property is a fundamental concept in algebra used to simplify expressions and perform multiplication over addition or subtraction.

In this context, it states that \(a(b + c) = ab + ac\). This property allows us to distribute a multiplication operation over an addition operation inside parentheses.

In the given problem,
  • Recognized in reverse as extracting the common factor.
  • We noticed \((y+2)\) as the common factor, hence using the distributive property in reverse to factor it out.
  • We transform the expression into \((y+2)(x+3)\), simplifying the expression into smaller parts.
This property is not only useful in factoring but also in expanding expressions and solving equations, making it a versatile tool for tackling algebraic problems.

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Most popular questions from this chapter

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ n(n+2)=360 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 20 x^{2}+41 x+20=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ n(n-24)=-128 $$

Problems \(63-100\) should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. (Objective 4) $$ 3 x y+15 x-2 y-10 $$

The perimeter of a rectangle is 32 inches, and the area is 60 square inches. Find the length and width of the rectangle.
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