Factor the expressions over the set of complex numbers. For assistance, consider these examples

Donald Johnson

Donald Johnson

Answered question

2022-01-15

Factor the expressions over the set of complex numbers. For assistance, consider these examples
In Section R.3 we saw that some expressions factor over the set of integers. For example: x24=(x+2)(x2)
Some expressions factor over the set of irrational numbers. For example: x25=(x+5)(x5)
To factor an expression such as x2+4, we need to factor over the set of complex numbers. For example, verify that x2+4=(x+2i)(x2i)
a) x29, x2+9
b) x2100, x2+100
c) x264, x2+64

Answer & Explanation

eskalopit

eskalopit

Beginner2022-01-16Added 31 answers

Step 1
Consider, right hand side of the given expression,
RHS =(x+2i)(x2i)
=x22ξ+2ξ4i2
=x2+04(1)
=x2+4
=LHS
PHS=LHS
Step 2
Therefore the given expression is true for factor over the complex numbers.
a) x29x29=(x3)(x+3)
x2+9x2+9=(x+3i)(x3i)
b) x2100x2100=(x+10)(x10)
x2+100x2+100=(x+10i)(x10i)
c) x264x264=(x+8)(x8)
x2+64x2+64=(x+8i)(x8i)

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