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Chapter 5: Problem 52

Factor. $$ x(x-y-z)+y(x-y-z) $$

Short Answer

Expert verified
The factored form is \((x-y-z)(x+y)\).

Step by step solution

01

Identify Common Factors

Observe the given expression \( x(x-y-z) + y(x-y-z) \). Notice that both terms in the sum have a common factor, which is \((x-y-z)\). This is the greatest common factor for this expression.
02

Factor Out the Common Factor

Since \((x-y-z)\) is a common factor in both terms, we can factor it out of the expression. The expression becomes: \[x(x-y-z) + y(x-y-z) = (x-y-z)(x+y) \]

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Key Concepts

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

Greatest Common Factor
The greatest common factor, often abbreviated as GCF, is a fundamental concept in algebra. It refers to the largest factor that two or more terms share. When factoring expressions, identifying the GCF can significantly simplify the process. In our example, the expression given is \( x(x-y-z) + y(x-y-z) \). Both parts of this expression share a common part, namely \((x-y-z)\). This commonality is the GCF in this case. By recognizing \((x-y-z)\) as the GCF, we can factor it out in the next steps, simplifying the entire expression considerably.
Polynomial Expressions
Polynomial expressions are algebraic expressions made up of terms, each consisting of a variable raised to a power and multiplied by a coefficient. In our exercise, we are given a binomial, which is a type of polynomial expression: \( x(x-y-z) + y(x-y-z) \). Understanding the nature of polynomial expressions allows us to manipulate and rewrite them in a more simplified form through factoring. By breaking down polynomials into factors, especially identifying common factors, solving equations becomes much easier and less error-prone. Recognizing the structure of polynomial expressions is crucial when simplifying or rewriting them in different forms.
Factoring Techniques
Factoring is a powerful algebraic tool used to simplify polynomial expressions. One of the most common techniques is factoring by grouping, where you identify and extract a common factor from the terms in an expression. In our given example, we see the expression \( x(x-y-z) + y(x-y-z) \). By recognizing \((x-y-z)\) as a common factor, we can employ factoring by grouping. This technique transforms the expression to \((x-y-z)(x+y)\). Factoring techniques are essential for simplifying expressions, solving polynomial equations, and making calculations more manageable in algebra.

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Most popular questions from this chapter

Factor. Assume all variables represent natural numbers. $$ 27 x^{3 n}+y^{3 n} $$

Explain why the factorization is not complete. $$ \text { Factor: } \quad 1-t^{8}=\left(1+t^{4}\right)\left(1-t^{4}\right) $$

Describe each expression in words:$$ (x+y)^{2} \quad(x-y)^{2} \quad(x+y)(x-y) $$

Factor each expression. $$ 1-y^{6} $$

Calculus. In the advanced mathematics course called Calculus, an important polynomial function is $$f(x)=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}$$ Refer to the polynomial on the right side of the equation. A. How many terms does the polynomial have? B. What is the degree of the polynomial? C. What is the coefficient of the fourth term? D. Find \(f(1)\)
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