What technique confirms the correctness of factored expressions?

What technique confirms the correctness of factored expressions?

• A. Substitution
• B. Mental calculation
• C. Re-expansion
• D. Estimation

Answer: (C)

Re-expansion is the technique that confirms the correctness of factored expressions by taking the factored form and expanding it back into its polynomial form. When you re-expand, you multiply the factors together, which allows you to check if you obtain the original expression you started with. This technique is particularly useful when verifying whether the factored expression has been factored correctly. For example, if you have factored a polynomial into the product of two binomials, re-expansion involves applying the distributive property to ensure that you arrive back at the original polynomial. If the two forms match, it confirms that the factoring was done correctly. This method is straightforward and provides a clear verification step in the process of working with polynomials. Other techniques mentioned, such as substitution, mental calculation, and estimation, serve different purposes. Substitution typically is used to test specific values within an equation, mental calculation is more about performing arithmetic operations without written work, and estimation helps in finding approximate values rather than verifying the exactness of expressions. These methods do not specifically focus on the process of confirming the accuracy of factored expressions like re-expansion does.

Re-expansion is the technique that confirms the correctness of factored expressions by taking the factored form and expanding it back into its polynomial form. When you re-expand, you multiply the factors together, which allows you to check if you obtain the original expression you started with. This technique is particularly useful when verifying whether the factored expression has been factored correctly.

For example, if you have factored a polynomial into the product of two binomials, re-expansion involves applying the distributive property to ensure that you arrive back at the original polynomial. If the two forms match, it confirms that the factoring was done correctly. This method is straightforward and provides a clear verification step in the process of working with polynomials.

Other techniques mentioned, such as substitution, mental calculation, and estimation, serve different purposes. Substitution typically is used to test specific values within an equation, mental calculation is more about performing arithmetic operations without written work, and estimation helps in finding approximate values rather than verifying the exactness of expressions. These methods do not specifically focus on the process of confirming the accuracy of factored expressions like re-expansion does.