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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 34
Chapter 1, Problem 34

Factor each polynomial by grouping. See Example 2. 20z2−8x+5pz2−2px20z^2-8x+5pz^2-2px

Verified step by step guidance
1
First, group the terms in pairs to prepare for factoring by grouping: \( (20z^{2} - 8x) + (5pz^{2} - 2px) \).
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 \(p\): \(p(5z^{2} - 2x)\).
Now, observe that both groups contain the common binomial factor \( (5z^{2} - 2x) \).
Factor out the common binomial factor \( (5z^{2} - 2x) \) from the entire expression: \( (5z^{2} - 2x)(4 + p) \).
The polynomial is now factored by grouping as \( (5z^{2} - 2x)(4 + p) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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