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Chapter 6: Problem 69

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

Short Answer

Expert verified
\((p + 4)(p + q)\)

Step by step solution

01

Group the terms

Break the expression into two groups: ingroup terms that can be factored together:\[ (p^2 + 4p) + (pq + 4q) \]
02

Factor out the common factor in each group

In the first group, \(p(p + 4)\) is factored out from \(p^2 + 4p\). In the second group, \(q(p + 4)\) is factored out from \(pq + 4q\):\[ p(p + 4) + q(p + 4) \]
03

Factor out the common binomial factor

Factor \( (p + 4) \) out 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.

Factoring Polynomials
Factoring polynomials is a fundamental skill in algebra, helping simplify expressions and solve equations. Polynomials are algebraic expressions made up of terms added together. When factoring polynomials, the goal is to break them down into simpler 'factorable' components. This can often make solving equations much easier.
Consider the polynomial in the exercise: \( p^{2} + 4p + pq + 4q \). We can rewrite this expression in a group-friendly way: \( (p^2 + 4p) + (pq + 4q) \).
Always remember that each group should allow for the extraction of a common factor. This leads to simpler polynomials.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions represent a value that can change depending on the values of the variables. Algebraic expressions form the basis of algebra.
In our exercise, we see an example of an algebraic expression: \( p^{2} + 4p + pq + 4q \). When dealing with algebraic expressions, we often need to simplify or rewrite them in different forms. Factoring by grouping is one way to simplify algebraic expressions. It involves pairing up terms and factoring out the greatest common factor (GCF) from each group. This way, we can make the expression even simpler.
Common Factors
Finding common factors is a key step in many algebra problems. A common factor is a number or algebraic term that divides each term in an expression without leaving a remainder. By identifying common factors, you can simplify expressions and solve equations more easily.
Let's look at our grouped expression: \( (p^2 + 4p) + (pq + 4q) \). To simplify it, we factor out the common factors:
  • From \( p^2 + 4p \), we can take out \( p \) to get \( p(p + 4) \)
  • From \( pq + 4q \), we can take out \( q \) to get \( q(p + 4) \)
Now we have \( p(p+4) + q(p+4) \). Notice \( (p+4) \) is a common binomial factor. Extracting this, we get: \( (p+4)(p+q) \). Recognizing and extracting common factors streamlines the process and simplifies the expression.

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Most popular questions from this chapter

Factor each trinomial completely. $$ 4 z^{2}-12 z w+9 w^{2} $$

Find each product. $$ (3 a+2)(2 a+1) $$

Factor each trinomial completely. $$ 64 x^{2}+48 x y+9 y^{2} $$

Find the value of the indicated variable. Find \(b\) so that \(100 a^{2}+b a+9\) factors as \((10 a+3)^{2}\)

Factor by grouping. \(10 t^{3}-2 t^{2} s^{2}-5 t s+s^{3}\)
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