Factor each polynomial by grouping. See Example 2. $20z^2-8x+5p{z}^2-2px$

Verified step by step guidance

1. First, group the terms in pairs to prepare for factoring by grouping: \( (20z^{2} - 8x) + (5pz^{2} - 2px) \).
2. Next, factor out the greatest common factor (GCF) from each group separately. From the first group \$20z^{2} - 8x\(, the GCF is 4, so factor out 4: \)4(5z^{2} - 2x)\(. From the second group \)5pz^{2} - 2px\(, the GCF is \)p\(, so factor out \)p\(: \)p(5z^{2} - 2x)$.
3. Now, observe that both groups contain the common binomial factor \( (5z^{2} - 2x) \).
4. Factor out the common binomial factor \( (5z^{2} - 2x) \) from the entire expression: \( (5z^{2} - 2x)(4 + p) \).
5. The polynomial is now factored by grouping as \( (5z^{2} - 2x)(4 + p) \).

Key Concepts

Polynomial Grouping

Polynomial grouping is a factoring technique where terms in a polynomial are grouped into pairs or sets that have common factors. By grouping terms strategically, you can factor out the greatest common factor from each group, simplifying the polynomial into a product of binomials or other factors.

Greatest Common Factor (GCF)

The greatest common factor is the largest expression that divides two or more terms without leaving a remainder. Identifying the GCF in each group of terms is essential for factoring by grouping, as it allows you to factor out common elements and simplify the polynomial.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps solve equations, simplify expressions, and analyze functions. Factoring by grouping is one method used when a polynomial has four or more terms.