Simplify the following expression.
left-parenthesis negative 4 plus 2 i right-parenthesis divided by left-parenthesis 1 plus i right-parenthesis
(1 point)
Responses
1 minus i
Image with alt text: 1 minus i
3 minus i
Image with alt text: 3 minus i
2 minus 2 i
Image with alt text: 2 minus 2 i
6 minus 2 i
To simplify the expression
(-4 + 2i) / (1 + i),
you can multiply the numerator and denominator by the conjugate of the denominator, which is (1 - i):
((-4 + 2i) / (1 + i)) * ((1 - i) / (1 - i))
Simplifying this expression further gives:
((-4 + 2i)(1 - i)) / ((1 + i)(1 - i))
= ((-4 + 4i - 2i - 2i^2)) / (1 - i^2)
Since i^2 equals -1, we can substitute its value:
= (-4 + 6i - 2(-1)) / (1 - (-1))
= (-4 + 6i + 2) / (1 + 1)
= (-2 + 6i) / 2
= -1 + 3i.
Therefore, the simplified expression is -1 + 3i.
To simplify the expression (−4 + 2i) / (1 + i), you can rationalize the denominator by multiplying the expression by the conjugate of the denominator.
The conjugate of 1 + i is 1 - i.
So, we can rewrite the expression as:
((-4 + 2i) / (1 + i)) * ((1 - i) / (1 - i))
Now, let's multiply the numerators and denominators:
(-4 + 2i)(1 - i) / (1 + i)(1 - i)
Expanding the numerator and denominator:
(-4 + 4i + 2i - 2i^2) / (1 - i + i - i^2)
Simplifying:
(-4 + 6i - 2i^2) / (1 - i^2)
Since i^2 = -1, we can substitute:
(-4 + 6i - 2(-1)) / (1 - (-1))
Simplifying further:
(-4 + 6i + 2) / (1 + 1)
Combining like terms:
(-2 + 6i) / 2
Now, divide each term by 2:
-2/2 + (6i)/2
Simplifying further:
-1 + 3i
Therefore, the simplified expression is -1 + 3i.