Final answer:
To factor the binomial 25xy^2 - 49x completely, we can factor out the greatest common factor and then apply the difference of squares formula.
Step-by-step explanation:
To factor the binomial 25xy^2 - 49x completely, we can look for the greatest common factor (GCF) of the two terms. In this case, the GCF is 1x, so we can factor it out:
25xy^2 - 49x = x(25y^2 - 49)
Next, we can simplify the expression inside the parentheses by recognizing that 25 and 49 are perfect squares. The difference between two perfect squares can be factored as the product of the sum and difference of their square roots:
25y^2 - 49 = (5y)^2 - 7^2 = (5y + 7)(5y - 7)
Combining these results, we have:
25xy^2 - 49x = x(5y + 7)(5y - 7)