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## Precalculus

### Course: Precalculus>Unit 3

Lesson 4: Identities with complex numbers

# Factoring polynomials using complex numbers

Sal shows how to factor a fourth degree polynomial into linear factors using the sum-product rule and the sum of squares identity. Created by Sal Khan.

## Want to join the conversation?

• The FIRST mistake is in writing out the problem. The polynomial given in the problem is x^4 + 5x^2 + 4. But the polynomial that Amat factored is x^4 + 10x^2 + 9.
• how did amat in the question above came up from x^4+5x^2+4 to x^4+10x^2+9
• They just made a mistake in the actaully problem itself. It was really meant to be x^4+10x^2+9, but somehow they accidentally wrote x^4+5x^2+4.
• Why is it (x +3i) and (x-3i) instead of (x+3) (x-3) for step 4?
• When you are using the zero product property, set each part equal to 0. So x^2 + 9 = 0 gives x^2 = -9, and there is no real number that can be squared to give a negative answer, so we have to go into the imaginary numbers. To get your answer, you need a difference of perfect squares (x^2 - 9).
• Hello, can you explain this part? how is this correct?
x^2 + a^2 = (x +ai)(x-ai)

Thank youu
(1 vote)
• (x + ai) * (x - ai)
= x * (x - ai) + ai * (x - ai)
= x^2 - xai + xai - (ai)^2
= x^2 - (ai)^2
= x^2 - (a^2 * i^2)
= x^2 - (a^2 * -1)
= x^2 + a^2