Factor each polynomial by grouping. See Example 2. 20z^2-8x+5pz^2-2px

Verified step by step guidance

1. Group the terms in pairs: \((20z^2 - 8x) + (5pz^2 - 2px)\).
2. Factor out the greatest common factor from each pair: \(4z^2(5) - 8x(1) + 5pz^2(1) - 2px(1)\).
3. Notice that the expression can be rewritten as \(4z^2(5) - 8x + 5pz^2 - 2px\).
4. Factor out the common binomial factor from the expression: \((4z^2 + 5pz^2) - (8x + 2px)\).
5. Combine the factored terms: \((4z^2 + 5pz^2) - (8x + 2px)\) to get the final factored form.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. This technique involves rearranging the terms into groups, factoring out the common factors from each group, and then factoring out the common binomial factor. It is particularly useful when the polynomial does not have a straightforward factorization.

Common Factors

A common factor is a number or variable that divides two or more terms without leaving a remainder. Identifying common factors is essential in factoring polynomials, as it allows for simplification of expressions. In the context of grouping, recognizing the common factors in each group is crucial for successfully factoring the polynomial.

Polynomial Terms

Polynomial terms are the individual components of a polynomial expression, consisting of coefficients and variables raised to non-negative integer powers. Understanding how to identify and manipulate these terms is fundamental in algebra. In the given polynomial, recognizing the structure and relationships between the terms is key to applying the grouping method effectively.