How to Factor 4x² + 25 Using Complex Numbers

Learn why 4x² + 25 cannot be factored over real numbers and how to factor it using complex numbers with imaginary unit i.

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The factored form of 4x^2 + 25 is not factorable using real numbers because it represents a sum of squares. However, in complex numbers, it can be expressed as (2x + 5i)(2x - 5i), where i is the imaginary unit satisfying i² = -1. This shows that while some expressions may not factor in the real number system, there are advanced concepts in algebra that allow for further factorization.

FAQs & Answers

  1. Why can't 4x² + 25 be factored using real numbers? 4x² + 25 is a sum of squares, and sums of squares do not factor into real linear factors. They can only be factored using complex numbers.
  2. What is the factored form of 4x² + 25 in complex numbers? Using complex numbers, 4x² + 25 factors as (2x + 5i)(2x - 5i), where i is the imaginary unit with i² = -1.
  3. What is the imaginary unit i in algebra? The imaginary unit i is defined such that i² = -1, and it allows for factorization of expressions that have no real factors.

Watch More

For a deeper understanding of polynomial factorization techniques, explore videos on factoring difference of squares, quadratic trinomials, and solving polynomials with complex roots.

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