Factor each polynomial completely. See Example 6. 8x³y⁴ + 12x²y³ + 36xy⁴

Verified step by step guidance

1. Identify the greatest common factor (GCF) of all the terms in the polynomial. Look at the coefficients (8, 12, 36), the powers of x (x³, x², x¹), and the powers of y (y⁴, y³, y⁴).
2. Factor out the GCF from each term. For the coefficients, find the largest number that divides 8, 12, and 36. For the variables, take the lowest power of x and y that appears in all terms.
3. Rewrite the polynomial as the product of the GCF and the remaining polynomial inside parentheses. This means expressing the original polynomial as \(\text{GCF} \times (\text{remaining terms})\).
4. Simplify the terms inside the parentheses by dividing each original term by the GCF you factored out.
5. Check if the polynomial inside the parentheses can be factored further. If it can, continue factoring; if not, the factorization is complete.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides all terms of a polynomial without leaving a remainder. Identifying the GCF helps simplify polynomials by factoring it out, making the remaining expression easier to work with.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or monomials. This process often starts by extracting the GCF, followed by factoring special products or applying methods like grouping to break down the expression completely.

Exponents and Variables in Factoring

When factoring terms with variables and exponents, it is important to consider the lowest power of each variable common to all terms. Factoring out variables involves subtracting exponents according to the laws of exponents, ensuring the factored form is accurate and simplified.