Complex Numbers and Factoring

The video tutorial explains how to factor expressions involving sums of squares using complex numbers. It introduces the concept that a^2 + b^2 can be factored as (a + bi)(a - bi) when using complex numbers. An example is provided, demonstrating the factoring of x^2 + 25 into (x + 5i)(x - 5i). The tutorial also verifies the correctness of this factorization by multiplying the factors and simplifying the expression.

What is a common misconception about factoring sums of squares?

• They can be factored using complex numbers.

• They cannot be factored at all.

• They can only be factored if they are differences of squares.

• They can only be factored using real numbers.

Which of the following is the correct factorization of x^2 + 25 using complex numbers?

• (x + 5)(x - 5)

• (x + 5i)(x - 5i)

• (x + 25i)(x - 25i)

• (x + 5)(x + 5)

In the example of factoring x^2 + 25, what are the values of a and b?

• a = x, b = 5

• a = x, b = 25

• a = 5, b = x

• a = 25, b = x

What happens to the middle terms when verifying the factorization of x^2 + 25?

• They simplify to x^2.

• They add up to zero.

• They remain unchanged.

• They cancel each other out.

What is the final simplified form of the expression after verifying the factorization of x^2 + 25?

• x^2 + 25i^2

• x^2 - 25i^2

• x^2 + 25

• x^2 - 25