Use DeMoivre's Theorem to determine the following power of a complex number. Write the answer in form a + bi, where a and b are real numbers and do not involve the use of a trigonometric function. (* - 4) - 5 V2

Q: 4. Given that 3-2i is a root of the equation z-6z³+18z²+pz+65-0, find all the roots and the real…

A: If  is a root of  then its conjugate  is also a root of  and hence  is a factor of .

Q: Show that the roots of px2 + (p - q)x - q = 0 are real for all real values of p and q.

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Q: If the roots of the equation x - 8x + a² - 6a = distinct, then find all possible values of a.

A: Since the roots of the given equation are real and distinct, We must have…

Q: What does Descartes’ Rule of Signs tell you about the real roots of x3 – x2 + 1 = 0 ?

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Q: Determine all real roots of x4 +0.9x³-2.3x²+3.6x-25.2=0

A: We need to find all the real roots of :

Q: Find all complex roots of the equation z6+z5+z4+z3+z2+z+1=0.

A: The given equation is z6+z5+z4+z3+z2+z+1=0    .....(1) We know that  z7-1=(z-1)(z6+z5+z4+z3+z2+z+1)…

Q: Find the value of the real number a if -1 – 2i is a root of the equation z² + az +5 =0. What is the…

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Q: Form a quadratic with real coefficients having amongst its roots -6+i

A: As given in this question, one root is imaginary then other will also be an imaginary roots.

Q: Find the four roots of ( 3+14)

A: The given problem is related with complex analysis. We have to find the four roots of 3 + i4. Note…

Q: 2 Express ( √5 - √5₂) in 2 simplest a + bi form.

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Q: Find the distinct roots of (-7 + 2j)3/4.

A: We know that, 1.Let z =rcosθ +isinθ    then zn =rncosnθ +isinnθ 2. A number u is said to be an n-th…

Q: For what values of the parameter a does the equation x + 2ax³ + x² + 2ax+1=0 have at least two…

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Q: If the roots of ax² + bx - 2 = 0 are a + b = 1+√3 3 I what is a+b?

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Q: Find the distinct roots of (-7 + 2i)3/4

A: Let x= (-7 + 2i)3/4 ⇒x4=-7+2i3 On Simplifying, ⇒x4=-259+286i--(1) Let, x=u+iv Now,…

Q: Find p so that  x2 - 6x + p = 0  has one root (i.e. is a perfect square)

A: We will solve it by two methods Shortcut method to find p for the equation given is x2 + b x + c  is…

Q: Find the four roots of the number 16- as a result of the assumption

A: To find fourth root of 16 by asumption. Let x=164

Q: Use DeMoivre’s theorem to find the cube roots of 8 (i.e. let’s solve the equation z3 = 8 for z).

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Q: Prove that the roots of ax+4=2x²-bx are real for all values of a and b

A: The nature of roots of a quadratic equation Ax2+Bx+C=0  is: