Factor by any method. See Examples 1–7. p⁴(m-2n)+q(m-2n)

Verified step by step guidance

1. Identify the common factor in both terms of the expression \(p^4(m-2n) + q(m-2n)\). Notice that \((m-2n)\) appears in both terms.
2. Factor out the common binomial factor \((m-2n)\) from the expression. This gives you \((m-2n)(p^4 + q)\).
3. Check the remaining factor inside the parentheses, \(p^4 + q\), to see if it can be factored further. Since \(p^4 + q\) is a sum of terms with no common factors and no special factoring formulas apply, it remains as is.
4. Write the fully factored form as the product of the common factor and the remaining expression: \((m-2n)(p^4 + q)\).
5. Verify your factorization by expanding the product to ensure it matches the original expression.

Key Concepts

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms in an expression to find common factors within each group. This method helps simplify complex polynomials by breaking them into smaller parts that share common factors, making it easier to factor the entire expression.

Common Factor Extraction

Common factor extraction is the process of identifying and factoring out the greatest common factor (GCF) from all terms in an expression. This simplifies the expression and is often the first step in factoring polynomials, as seen when terms share a binomial like (m - 2n).

Polynomial Expressions and Exponents

Understanding polynomial expressions and the role of exponents is essential for factoring. Recognizing powers, such as p^4, helps in identifying common bases and applying exponent rules, which is crucial when factoring terms with variables raised to powers.