Factor 2x^2y-4xy+6xy^2

Solution:

Step 1 Extract the common factor $2xy$ from the term $2x^{2}y$. This results in $2xy(x)-4xy+6x{y}^2$.

Step 2 Remove the common factor $2xy$ from the term $-4xy$. The expression now becomes $2xy(x)+2xy(-2)+6x{y}^2$.

Step 3 Take out the common factor $2xy$ from the term $6x{y}^2$. The equation simplifies to $2xy(x)+2xy(-2)+2xy(3y)$.

Step 4 Combine the terms with the common factor $2xy$. This gives us $2xy(x-2)+2xy(3y)$.

Step 5 Finally, factor out $2xy$ from the entire expression to get $2xy(x-2+3y)$.

Knowledge Notes:

Factoring is the process of finding an equivalent expression that is a product of simpler or smaller expressions. In the context of algebra, factoring polynomials involves expressing the polynomial as a product of its factors, which may include numbers, variables, or other polynomials.

The steps involved in the problem-solving process for factoring the given polynomial $2x^{2}y-4xy+6x{y}^2$ are as follows:

1. Identify the common factor across all terms: In this case, each term of the polynomial has a common factor of $2xy$.

2. Factor out the common factor: This involves dividing each term by the common factor and writing the polynomial as the product of the common factor and the resulting simplified terms.

3. Simplify the expression: After factoring out the common factor, the terms inside the parentheses are simplified to their lowest terms.

4. Combine like terms: If there are like terms within the parentheses after factoring, they should be combined to simplify the expression further.

5. Write the final factored form: The polynomial is now expressed as the product of the common factor and the simplified expression within the parentheses.

In the given problem, the final factored form is $2xy(x-2+3y)$, which is the product of the common factor $2xy$ and the binomial $x-2+3y$.