Divide the following complex numbers. (4+2i)/(-1+i)

Multiply by Conjugate: To divide the complex numbers (4+2i) and (-1+i), we need to multiply the numerator and denominator by the conjugate of the denominator to remove the imaginary part from the denominator. The conjugate of (-1+i) is (-1-i). So, we multiply both the numerator and the denominator by (-1-i). (4+2i)/(-1+i) * (-1-i)/(-1-i)Distribute Numerator: Now, we distribute the numerator: (4+2i) * (-1-i) = 4*(-1) + 4*(-i) + 2i*(-1) + 2i*(-i) = -4 - 4i - 2i + 2i^2 Since i^2 = -1, we replace 2i^2 with -2. = -4 - 4i - 2i - 2 = -6 - 6iDistribute Denominator: Next, we distribute the denominator: (-1+i) * (-1-i) = (-1)*(-1) + (-1)*(-i) + i*(-1) + i*(-i) = 1 + i - i - i^2 Again, since i^2 = -1, we replace -i^2 with 1. = 1 + i - i + 1 = 2Simplify Result: Now we have the simplified numerator and denominator: Numerator: -6 - 6i Denominator: 2 We divide both the real part and the imaginary part of the numerator by the denominator: (-6 - 6i) / 2 = -6/2 - (6i/2) = -3 - 3i

Home

Math Problems

Algebra 2

Add, subtract, multiply, and divide polynomials

Full solution

Q. Divide the following complex numbers.

\newline

\frac{4+2 i}{-1+i}

Multiply by Conjugate: To divide the complex numbers $(4+2i)$ and $(-1+i)$ , we need to multiply the numerator and denominator by the conjugate of the denominator to remove the imaginary part from the denominator. $\newline$ The conjugate of $(-1+i)$ is $(-1-i)$ . $\newline$ So, we multiply both the numerator and the denominator by $(-1-i)$ . $\newline$ $\frac{4+2i}{-1+i} \cdot \frac{-1-i}{-1-i}$
Distribute Numerator: Now, we distribute the numerator: $\newline$ $(4+2i) \times (-1-i) = 4\times(-1) + 4\times(-i) + 2i\times(-1) + 2i\times(-i)$ $\newline$ $= -4 - 4i - 2i + 2i^2$ $\newline$ Since $i^2 = -1$ , we replace $2i^2$ with $-2$ . $\newline$ $= -4 - 4i - 2i - 2$ $\newline$ $= -6 - 6i$
Distribute Denominator: Next, we distribute the denominator: $\newline$ $(-1+i) \cdot (-1-i) = (-1)\cdot(-1) + (-1)\cdot(-i) + i\cdot(-1) + i\cdot(-i)$ $\newline$ $= 1 + i - i - i^2$ $\newline$ Again, since $i^2 = -1$ , we replace $-i^2$ with $1$ . $\newline$ $= 1 + i - i + 1$ $\newline$ $= 2$
Simplify Result: Now we have the simplified numerator and denominator: $\newline$ Numerator: $-6 - 6i$ $\newline$ Denominator: $2$ $\newline$ We divide both the real part and the imaginary part of the numerator by the denominator: $\newline$ $(-6 - 6i) / 2$ $\newline$ $= -6/2 - (6i/2)$ $\newline$ $= -3 - 3i$