cubic equation with complex root?

Question

Given that (x - 2) is a factor of the expression x³ + ax² - 4, find the value of a. Hence, solve the equation x³ + ax² - 4 = 0, expressing the complex root in the form of a + bi

Answer 1

Let's call the expression p(x).
p(x) = x^3 + ax^2 - 4.

If (x - 2) is a factor of the expression x^3 + ax^2 - 4, then that mean p(2) = 0. However, p(2) is also equal to what is stated below:

p(2) = 2^3 + a(2)^2 - 4

Equating the above to 0, we get
2^3 + a(2)^2 - 4 = 0. Simplifying this, we get
8 + 4a - 4 = 0
4a = -4
a = -1

Therefore, the value of a = -1, and our expression is

p(x) = x^3 - x^2 - 4

Since we *know* that (x - 2) is a factor, we can perform long division on p(x). I won't show you the details of the actual synthetic long division on here (since it's difficult to show), but I can tell you the result of the division should be x^2 + x + 2.

This means that p(x) = (x - 2) (x^2 + x + 2).

Equating this to 0 means we equate each factor to 0, i.e.
x - 2 = 0 (which means x = 2), and
x^2 + x + 2 = 0 (which means we use the quadratic formula to determine the results).

x^2 + x + 2 = 0 means
x = [-1 +/- sqrt(1 - 4(1)(2))]/2
x = [-1 +/- sqrt(-7)]/2

But, sqrt(-7) = sqrt(-1 * 7) = (i) sqrt(7), so

x = [-1 +/- i sqrt(7)]/2
x = (-1/2) +/- [(1/2)sqrt(7)]i

Thus, our three solutions are
x = {2, (-1/2) + [(1/2)sqrt(7)]i , (-1/2) - [(1/2)sqrt(7)]i }

Answer 2

Since (x - 2) is a factor of the expression x³ + ax² - 4, it means that 2 is a root of the equation x³ + ax² - 4=0.

x³ + ax² - 4=0

2^3 + a(2^2) - 4 = 0

8 +4a - 4 = 0

a = -1.

To get the other roots:

x^3 -x^2 - 4 = (x-2)(x^2 + x + 2)= 0

The other factor was obtained by dividing x^3 -x^2 - 4 by x-2

x^2+x+2 = 0

Use the quadratic formula to get the values of x.

x = [-1 + sqrt (1^1 - 4(1)(2) ] / 2(1)

x = [-1 - sqrt (1^1 - 4(1)(2) ] / 2(1)

Answer 3

Let f(x) = x^3 + ax^2 - 4.
If one divides f(x) by x - 2, then we obtain
g(x) = x^2 + (a+2)x + 2(a+2) as the quotient (if a is not -2) and
h(x) = 4(a+2) - 4 as the remainder.
Since x - 2 is a factor of f(x), the remainder should be zero.
Hence, a = -1. Thus, the quotient g(x) = x^2 + x + 2.
Using the quadratic formula, we find the complex roots:
x = (-1 +- sqrt(7)i)/2 = -1/2 -+ sqrt(7)i/2.

Answer 4

Well from what I see you will have to divide the cubic expression by the factor. You can do this by either synthetic division or just standard long division.

This gives you a remainder of 4 + 4a, which has to equal 0 because (x-2) is a factor.

Solve for x, and you get a = -1

Now substitute a, and divide again, this time noting the quotient (the answer after you divide).

All you need to do now is solve for x, using the quadratic formula and simplify enough to get into a+bi form.

Answer 5

f(x) = x^3+ax^2-4

becuase (x-2) is a factor

f(2) = 8+4a - 4 = 0
or 4a = - 4
or a = -1

f(x) = x^3-x^2-4
= x^3 -2x^2 + x^2 - 4
= x^(x-2) + (x-2)(x+2)
= (x-2)(x^2+x+2)

now we need to slove

x^2+x+1 = 0 or (x+1/2)^2 + 7/4 = 0
solutions are
x = -1/2 +/- i sqrt(7)/2

Answer 6

it relatively is relatively a tremendously undemanding question to respond to. If 2 is between the roots of the cubic equation, you already know that when z=2, the function is comparable to 0. So, between the words of the equation is (z-2). because you already know that the roots of (z^2 + 6z + thirteen) are -3+2i and -3-2i, you presently have sufficient information to remedy the undertaking. If we multiply (z^2 + 6z + thirteen)(z-2), we are able to get the the suitable option function, it relatively is: z^3 + 6z^2 + 13z - 2z^2 - 12z - 26 = z^3 + 4z^2 + z - 26 in actuality, multiplying (z - root1)(z - root2)(z - root3) provide you the respond, and because root1 and root2 multiply to grant you z^2 + 6z + thirteen, all we ought to do is multiply via the final root to get the suitable answer.

Answer 7

okay....what grade? whatever.

i= a negative form of a factor which cannot be described as an integer ex: 2.

then a must be 4!
4 + (-b)=0?
hold on...i did something wrong. i think.

x-2 /x cubed plus ax2 -4....
yes, a =4. since x-2 divided by x3 +ax2 -4 is x2 +3x +6 remainder a=16, so factor of a is 4

substituting.....
x3+4x2-4
um....hey, i'm only 13, okay? its hard....i have to remember. and my answers are correct!

um...you have to start substituting for me....
i cant find it. its x= one of following: 2, -2, 4, -4, 1/4, -1/4, 1/2, or -1/2. i got you that far.