Problem 83 Factor by grouping. \(p^{2}+4 ... [FREE SOLUTION]

Chapter 5: Problem 83

Factor by grouping. \(p^{2}+4 p+p q+4 q\)

Short Answer

Expert verified

(p + 4)(p + q)

Step by step solution

- Group terms

Start by grouping the terms that have common factors. The expression is \(p^{2} + 4p + pq + 4q\). Group the terms as follows: \((p^{2} + 4p) + (pq + 4q)\).

- Factor out the common factor from each group

Factor out the greatest common factor (GCF) from each group separately: \(p(p + 4) + q(p + 4)\).

- Factor out the common binomial factor

Notice that \((p + 4)\) is common in both groups. Factor \((p + 4)\) from the entire expression: \((p + 4)(p + q)\).

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Key Concepts

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

Finding the Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that two or more numbers have in common. It's important in factoring because it helps us simplify algebraic expressions.

To find the GCF, you can follow these steps:

List the factors of each number or term.
Identify the common factors.
Select the largest of these common factors.

In our exercise, when we look at the first group \((p^2 + 4p)\), the GCF is \(p\). For the second group \((pq + 4q)\), the GCF is \(q\). Factoring these out gives us \p(p + 4) + q(p + 4)\.

Binomial Factorization

Binomial factorization involves expressing a polynomial as a product of binomials.

To do this, we look for common terms in each binomial group and factor them out.

If a common term exists in both groups, we can factor this term out to simplify our expression.

This concept is illustrated in our exercise where after we factored out the GCFs from each group, we noticed \((p + 4)\) appeared in both terms: \p(p + 4) + q(p + 4)\.

From here, we can factor out the binomial \((p + 4)\), simplifying to \((p + 4)(p + q)\).

Algebraic Expressions

Understanding algebraic expressions is crucial as they form the foundation of algebra.

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. For example, \p^2 + 4p + pq + 4q\ is an algebraic expression with terms.

Grouping terms is often the first step in simplifying complex algebraic expressions.
Factoring out common factors can further simplify these expressions.

In our exercise, we initially grouped the terms: \((p^2 + 4p) + (pq + 4q)\). Through these steps, we simplified the expression to: \((p + 4)(p + q)\).

This not only makes it easier to work with but also helps in solving equations and understanding the structure of expressions.

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91影视

Factor by grouping. \(p^{2}+4 p+p q+4 q\)

Short Answer

Step by step solution

- Group terms

- Factor out the common factor from each group

- Factor out the common binomial factor

Key Concepts

Finding the Greatest Common Factor

Binomial Factorization

Algebraic Expressions

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