Problem

Classify \(x^{2}+25\) over the integers. Identify the operation and square terms, test whether the difference-of-squares identity applies, and distinguish integer factorization from a possible complex factorization.

Big Picture

What this problem is really about

Perfect-square terms alone do not trigger the difference-of-squares identity; the operation between them matters. We’ll classify the expression as a sum, test the integer binomial conditions, and keep the requested integer result separate from the different factorization available over the complex numbers.

Answer choices

Choose your path.

Preview only

Operation: sum; square terms x² and 5²; difference-of-squares over integers: yes; integer factorization: (x − 5)(x + 5); field note: conjugates are used for a sum of squares.

Operation: sum; square terms x² and 25; difference-of-squares over integers: yes; integer factorization: (x − √25)(x + √25); field note: any positive constant can be treated as an integer square difference.

Operation: sum; square terms x² and 5²; difference-of-squares over integers: no; integer factorization: (x − 5i)(x + 5i); field note: a complex factorization is incorrectly labeled an integer factorization.

Operation: sum; square terms x² and 5²; difference-of-squares over integers: no; integer factorization: none; field note: a sum of nonzero squares does not use the difference identity.

The choices are visible. Checking your work and seeing which answer is correct happen inside Gozunta.

Classify \(x^{2}+25\) over the integers. Identify the operation and square terms, test whether the difference-of-squares identity applies, and distinguish integer factorization from a possible complex factorization.

What this problem is really about

Turn the preview into practice.

Gozunta lets you study this problem—and more than 10,000 others.