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Chapter 5: Problem 134

Factor each polynomial completely. $$w^{2}-y w-y+w$$

Short Answer

Expert verified
(w - y)(w - 1)

Step by step solution

01

- Group the Terms

Group the polynomial in pairs to make factoring easier. In this case, group the first two terms together and the last two terms together: \( w^{2} - yw + (-y + w) \)
02

- Factor Out the Greatest Common Factor (GCF) from Each Pair

Factor out the greatest common factor (GCF) from each pair of terms: \( w(w - y) - 1(w - y) \)
03

- Factor by Grouping

Since \( (w - y) \) is a common factor, factor it out to get the complete factorization: \( (w - y)(w - 1) \)

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Key Concepts

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

Grouping Terms
To start factoring the polynomial, we first need to look at grouping terms. This technique involves dividing the given polynomial into smaller groups to make the factorization process easier. In our example, we have the polynomial: \( w^{2} - y w - y + w \). To apply grouping, we need to group the terms in pairs, like this: \( (w^{2} - yw) + (-y + w) \). By grouping terms, we simplify the polynomial and make it easier to identify common factors.
Greatest Common Factor
Next, we need to find the Greatest Common Factor (GCF) for each group. The GCF is the highest factor that divides all terms in the group. For the first group, \( w^{2} - yw \), the GCF is \( w \) because both terms include a factor of \( w \). We can factor this out: \( w(w - y) \). For the second group, \( -y + w \), notice how there is no common factor other than 1, so we factor out \( -1 \): \( -1(y - w) \) (we rewrite \( -y + w \) as \( -(y - w) \)). Combining these results gives us: \( w(w - y) - 1(y - w) \). Identifying the GCF makes factoring easier by reducing the polynomial.
Factor by Grouping
In the next step, we will factor by grouping. This technique involves using the GCF identified in each group to factor out the common terms. We have: \( w(w - y) - 1(y - w) \). Let's rewrite \( -1(y - w) \) to make factoring simpler: \( -1(y - w) = -1(-w + y) = (w - y) \). This adjustment lets us factor out: \( (w - y)(w - 1) \). By grouping, we transform the polynomial into a product of simpler binomials, making it completely factored.
Complete Factorization
In this final stage, we ensure that the polynomial is fully factored. Remember, a polynomial is completely factored when it is expressed as a product of irreducible polynomials. After following the steps above, our polynomial: \( w^{2} - yw - y + w \) has been reduced to: \( (w - y)(w - 1) \). Each binomial is irreducible, confirming complete factorization. This process demonstrates the importance of systematic steps: grouping terms, finding the GCF, factoring by grouping, and ensuring complete factorization. It's essential to practice these steps for polynomials to become skilled in factoring any given polynomial completely.

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