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Chapter 3: Problem 49

For Problems \(25-50\), factor completely. $$ x(x+2)+5(x+2) $$

Short Answer

Expert verified
Factored form is \((x+2)(x+5)\).

Step by step solution

01

Identify Common Factor

Examine the given expression: x(x+2) + 5(x+2).Notice that both terms contain the common factor \((x+2)\).
02

Factor Out the Common Factor

Factor out the common factor \((x+2)\) from the expression: \[(x+2)(x) + (x+2)(5) = (x+2)(x + 5)\].

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Key Concepts

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

Common Factor
The concept of a common factor is fundamental in simplifying expressions in algebra. A common factor is a term that appears in every part of an expression, allowing it to be factored out and simplified.
Let's take a closer look at the expression from our exercise:
  • The expression is: \( x(x+2) + 5(x+2) \).
  • Notice, \(x+2\) appears in both terms of the expression.
  • This makes \(x+2\) the common factor.

This clever observation reveals that each term is actually composed of two parts: the factor common to both terms, and another part unique to each term. Finding the common factor allows us to combine terms more easily and reduce complexity in problem-solving.
Algebraic Expression
An algebraic expression is a combination of variables, numbers, and at least one operation (such as addition or multiplication). These expressions represent a wide array of mathematical scenarios, from simple equations to complex functions.
In this exercise, the algebraic expression given is \( x(x+2) + 5(x+2) \). This expression consists of two terms:
  • \( x(x+2) \)
  • \( 5(x+2) \)
Algebraic expressions are often factored to make problem-solving easier.
Factoring simplifies the expression by grouping terms with common factors of \(x+2\). This helps in easily identifying and solving further mathematical problems by reducing the terms to simpler forms.
Polynomials
Polynomials are a category of algebraic expressions that consist of variables raised to whole number powers and their coefficients. They can have one or multiple terms, and can vary from simple linear polynomials to more complex ones with higher degrees.
The given expression \( x(x+2) + 5(x+2) \) represents a polynomial. To factor this polynomial, recognizing patterns such as common factors is crucial.
  • The expression can be rewritten in factored form as \( (x+2)(x+5) \).
  • This transformation simplifies solving equations or evaluating expressions.
These skills with polynomials will continue to be useful as you advance in algebra, allowing you to approach and simplify a variety of mathematical expressions and challenges with confidence.

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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) $$ w^{2}-4 w=5 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 24 n^{2}-38 n+15=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ n^{2}+20 n+91=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ (x-5)(x+3)=9 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 3 t(t-4)=0 $$
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