calculate P(i) for the following polynomial
p(x)=x3+2x2-5x+8
Find | z | for each of the following complex numbers and represent them on a complex plane.
a. z=-3-4i
b. v=5-12i
c.u=7+24i
d.w=-9+40i
e.p=1+3i
f.t=-2i
g.n=4
h.m=0
i.s=(√3-i)2
I don't mind checking your work, but I am not going to do it.
i was looking for help ... not to solve my hw. -.-
To calculate P(i) for the given polynomial p(x) = x^3 + 2x^2 - 5x + 8, you need to substitute i for x in the polynomial and simplify:
P(i) = (i)^3 + 2(i)^2 - 5(i) + 8
To evaluate powers of i, recall that i is defined as the imaginary unit where i^2 = -1. Therefore, i^3 = i * i^2 = i * (-1) = -i.
Substituting these values back into the equation:
P(i) = -i + 2(-1) - 5i + 8
= -i - 2 - 5i + 8
= 6 - 6i
So, P(i) = 6 - 6i.
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Now, let's calculate |z| for each of the complex numbers and represent them on a complex plane:
a. z = -3 - 4i:
|z| = sqrt((-3)^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5
On the complex plane, z can be represented as the point (-3, -4).
b. v = 5 - 12i:
|v| = sqrt((5)^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13
On the complex plane, v can be represented as the point (5, -12).
c. u = 7 + 24i:
|u| = sqrt((7)^2 + (24)^2) = sqrt(49 + 576) = sqrt(625) = 25
On the complex plane, u can be represented as the point (7, 24).
d. w = -9 + 40i:
|w| = sqrt((-9)^2 + (40)^2) = sqrt(81 + 1600) = sqrt(1681) = 41
On the complex plane, w can be represented as the point (-9, 40).
e. p = 1 + 3i:
|p| = sqrt((1)^2 + (3)^2) = sqrt(1 + 9) = sqrt(10)
On the complex plane, p can be represented as the point (1, 3).
f. t = -2i:
|t| = sqrt((0)^2 + (-2)^2) = sqrt(0 + 4) = sqrt(4) = 2
On the complex plane, t can be represented as the point (0, -2).
g. n = 4:
|n| = sqrt((4)^2) = sqrt(16) = 4
On the complex plane, n can be represented as the point (4, 0).
h. m = 0:
|m| = sqrt((0)^2) = 0
On the complex plane, m can be represented at the origin (0, 0).
i. s = (√3 - i)^2:
Expanding (√3 - i)^2:
s = (√3 - i)(√3 - i) = (√3)^2 - √3i - √3i + i^2
= 3 - 2√3i - 1
= 2 - 2√3i
|s| = sqrt((2)^2 + (-2√3)^2) = sqrt(4 + 12*3) = sqrt(4 + 36) = sqrt(40) = 2√10
On the complex plane, s can be represented as the point (2, -2√3).
I hope this helps! Let me know if you have any further questions.