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Chapter 10: Problem 620

Factor the greatest common factor from each polynomial. $$6 p^{2}+6 p$$

Short Answer

Expert verified
The factored form is \(6p(p + 1)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Analyze the terms in the polynomial \(6p^2 + 6p\). Both terms contain the factor \(6p\). Identify \(6p\) as the greatest common factor (GCF).
02

Factor out the GCF

Rewrite the polynomial factoring out the GCF \(6p\). This means you divide each term by \(6p\) and place it outside of the parentheses: \(6p^2 + 6p = 6p(p + 1)\).

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Key Concepts

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

greatest common factor
Understanding the Greatest Common Factor (GCF) is essential before diving into polynomial factorization. The GCF of two or more numbers or algebraic expressions is the largest factor that divides each of them without leaving a remainder. To find the GCF of the polynomial \(6p^2 + 6p\), you need to look for the common factors in each term.

In this case, both terms (\(6p^2\) and \(6p\)) share the factor 6 and the variable p. Therefore, the GCF of the polynomial is 6p. Identifying the GCF helps simplify the polynomial and make further factorization more straightforward.
polynomial factorization
Polynomial factorization involves expressing a polynomial as a product of its factors, which are simpler polynomials or constants. It's like breaking down a complex expression into simpler parts. When you factor out the GCF from a polynomial, you are performing polynomial factorization.

Consider the polynomial \(6p^2 + 6p\). After identifying the GCF, which is \(6p\), you factor it out by dividing each term by \(6p\) and placing it outside the parentheses:
\[ 6p^2 + 6p = 6p(p + 1). \]
This shows that \(6p^2 + 6p\) can be broken down into the product of \(6p\) and \(p + 1\), making the expression simpler and more manageable.
algebraic expression
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In the context of the given polynomial exercise, the expression \(6p^2 + 6p\) is an algebraic expression containing terms with the variable \(p\).

Understanding how to manipulate these expressions, such as factoring, is a critical skill in algebra. Factorization helps simplify complex expressions, making them easier to work with and solve. By mastering the techniques of working with algebraic expressions, like identifying the GCF and factoring polynomials, students can tackle more advanced algebraic problems effectively.

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Most popular questions from this chapter

Simplify. $$5 \cdot 2^{-4}$$

Simplify each expression. $$\frac{x^{4}}{x^{-5}}$$

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Simplify each expression. $$(n-8)\left(n^{2}-4 n+11\right)$$

Find the greatest common factor. $$12 x^{2}, 20 x^{3}, 36 x^{4}$$
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