Final answer:
The expressions were factored by identifying the greatest common factor (GCF) and factoring it out of each term. The result was checked to ensure that the factored expression multiplied back equals the original expression.
Step-by-step explanation:
To factor the given expressions completely, we need to identify the greatest common factor (GCF) that can be factored out from each term of the expression. Below are the factored forms of the provided expressions:
- 10x^2 - 5x: The GCF is 5x. Factoring out 5x gives us 5x(2x - 1).
- 36x^2y^2 - 30xy: The GCF is 6xy. Factoring out 6xy gives us 6xy(6xy - 5).
- uvr + u^2y^2r^2 + u^3vr^3: The GCF is uvr. Factoring out uvr gives us uvr(1 + uyr + u^2r^2).
- 28abc - 14abc^3 + 7abc: The GCF is 7abc. Factoring out 7abc gives us 7abc(4 - 2c^2 + 1).
After factoring, it is essential to check if the result is reasonable. Each term in the original expression should be divisible by the GCF, and if you distribute the GCF back into the factored form, you should get the original expression as a product. Simplifying the algebra involves eliminating terms where possible and ensuring that the simplified form is mathematically equivalent to the original expression.