Factor out the greatest common factor from each polynomial. See Example 1. $-4p^3{q}^4-2p^2{q}^5$

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the coefficients. For -4 and -2, the GCF is 2, but since both terms are negative, the GCF will be -2.
2. Determine the GCF of the variable parts by looking at the powers of each variable. For \(p^3\) and \(p^2\), the GCF is \(p^2\) because it is the lowest power of \(p\). For \(q^4\) and \(q^5\), the GCF is \(q^4\) for the same reason.
3. Combine the GCF of the coefficients and variables to get the overall GCF: \(-2p^2q^4\).
4. Divide each term of the polynomial by the GCF \(-2p^2q^4\) to find the remaining factors inside the parentheses.
5. Write the factored form as the GCF multiplied by the simplified expression inside parentheses: \(-2p^2q^4(\text{simplified terms})\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest factor that divides two or more terms without leaving a remainder. In polynomials, it includes the highest power of variables and the largest numerical coefficient common to all terms. Factoring out the GCF simplifies the expression and is the first step in polynomial factorization.

Factoring Polynomials

Factoring polynomials involves rewriting the expression as a product of simpler polynomials or factors. Extracting the GCF is often the initial step, which reduces the polynomial to a simpler form, making further factoring or solving easier. This process helps in simplifying expressions and solving equations.

Exponents and Variables in Factoring

When factoring polynomials with variables, the GCF includes the variable raised to the lowest exponent present in all terms. Understanding how to compare and factor out variables with exponents is essential, as it ensures the correct common factors are extracted without altering the polynomial's value.