Problem 10 - College Algebra

Question

Factor out the greatest common factor. $$ x^{2}(2 x+5)+17(2 x+5) $$

Step-by-Step Solution

Answer

Factored form is \((2x + 5)(x^{2} + 17)\).

Step 1: Identify common factor

First, recognize that the common factor in the expression is \(2x + 5\). Both \(x^{2}(2 x+5)\) and \(17(2 x+5)\) share this piece.

Step 2: Factor out common factor

Next, factor the common term out from both terms. This leaves us with \(x^{2}\) from \(x^{2}(2 x+5)\) and \(17\) from \(17(2 x+5)\). Writing these together results in \((2x+5) \cdot (x^{2}+17)\).

Step 3: Write out final factored form

Writing this out, we get the factored form of our original expression: \((2x + 5)(x^{2} + 17)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an important concept in algebra. It is the largest factor that divides two or more numbers or expressions. In polynomials, finding the GCF helps to simplify expressions by "factoring out" shared components. This reduces the complexity and can make solving equations easier.

To find the GCF in an algebraic expression, look for terms that repeatedly appear in each component of the polynomial. For example, in the expression \(x^{2}(2x+5) + 17(2x+5)\), the term \(2x + 5\) is present in both parts of the expression.

Once you identify the GCF, you "factor it out". This means you divide each term by this common factor and multiply it by the factor as a whole. Factoring out the GCF simplifies the expression and is a crucial step in the process of factoring polynomials.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They are the building blocks in algebra, representing quantities without fixed values. Understanding how to manipulate these expressions is key to solving algebraic problems.

In our problem, the expression \(x^{2}(2x+5) + 17(2x+5)\) is an algebraic expression. It involves multiplication and addition, with variables \(x\) and their coefficients. These terms are combined using arithmetic operations to form an expression.

When working with algebraic expressions, one of the primary goals is to simplify them. This might involve combining like terms or factoring polynomials. Recognizing the structure and components of the expression makes it easier to manipulate and solve equations.

Factored Form

The factored form of an expression represents it as a product of its factors. Converting to factored form simplifies the expression and often reveals insights about its roots and solutions.

Consider our expression: \(x^{2}(2x+5) + 17(2x+5)\). Once the common term \(2x+5\) is factored out, we are left with the expression in its factored form: \((2x+5)(x^{2}+17)\).

Transforming into factored form is essential in solving equations and simplifying expressions. It allows us to efficiently work with equations by breaking them into more manageable parts. This can also make complex problems easier to solve and offer clarity towards the solution.