4
To simplify the expression ((7 - 3i)(-11 + 5i)), use the distributive property (FOIL method). First, multiply the real parts: (7 \cdot -11 = -77). Next, multiply the outer parts: (7 \cdot 5i = 35i), then the inner parts: (-3i \cdot -11 = 33i), and finally the last parts: (-3i \cdot 5i = -15i^2). Combine the results: (-77 + 35i + 33i + 15) (since (i^2 = -1)) simplifies to (-62 + 68i). Thus, the simplified expression is (-62 + 68i).
The four roots are:1 + 2i, 1 - 2i, 3i and -3i.
i is the Imaginary Unit, equal to sqrt(-1). So i and any real number multiplied by i will all be imaginary numbers. Here are some: i, -i, 5i, -3i, i*pi, etc.
(3x - 1)(x - 4) so roots are 1/3 and 4
1/(1+ 3i)
When adding and subtracting complex numbers, you can treat the "i" as any variable. For example, 5i + 3i = 8i, 5i -3i = 2i, etc.; (2 + 5i) - (3 - 3i) = (2 - 3) + (5 + 3)i = -1 + 8i.
2
There cannot be such a polynomial. If a polynomial has rational coefficients, then any complex roots must come in conjugate pairs. In this case the conjugate for 2-3i is not a root. Consequently, either (a) the function is not a polynomial, or (b) it does not have rational coefficients, or (c) 2 - 3i is not a root (nor any other complex number), or (d) there are other roots that have not been mentioned. In the last case, the polynomial could have any number of additional (unlisted) roots and is therefore indeterminate.
-4-3i
The four roots are:1 + 2i, 1 - 2i, 3i and -3i.
1
1/(2 + 5i) (multiply both the numerator and the denominator by 2 - 5i)= 1(2 - 5i)/(2 + 5i)(2 - 5i)= (2 - 5i)/(4 - 25i2) (substitute -1 for i2)= (2 - 5i)/(4 + 25)= (2 - 5i)/29= 2/29 - (5/29)i
1/(3+5i)=(3-5i)/((3+5i)(3-5i))=(3-5i)/(9+25)=(3-5i)/34
It can have 1, 2 or 3 unique roots.
i is the Imaginary Unit, equal to sqrt(-1). So i and any real number multiplied by i will all be imaginary numbers. Here are some: i, -i, 5i, -3i, i*pi, etc.
(3x - 1)(x - 4) so roots are 1/3 and 4
The 3 cube roots of a cubic equation are often going to require complex numbers, and this is one of them. The primary root is (-2) since (-2)3= -8 The other two roots are going to have the same mod (2) but be evenly spaced around the complex plane, so (2pi)/3 angular difference.These angles are pi/3 and 5pi/3(2,pi/3) = (1 + √3i) = (1 + 1.732i)(2,5pi/3) = (1 + -√3i) = (1- 1.732i)Check(-2)3 = -8◄(1 + √3i)3 = -8◄(1 - √3i)3 = -8◄■