SOLUTION: Q. Show that 2 - i is a root of z^4 - 8z^3 + 28z^2

SOLUTION: Q. Show that 2 - i is a root of z^4 - 8z^3 + 28z^2 - 48z + 35. Hence fond all the roots

Question 135973: Q. Show that 2 - i is a root of z^4 - 8z^3 + 28z^2 - 48z + 35. Hence fond all the roots
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Actually, technically speaking, and we should always speak technically when talking about this stuff, doesn't have any roots. has roots, and those roots are related to the factors of as we shall see shortly.

Assume that is a root of

Then must also be a root of because complex roots always occur in conjugate pairs.

is a root of a polynomial equation if and only if is a factor of the polynomial. That means that both and are factors of . Furthermore, the product of those two factors must also be a factor of .

The product of and , works out to (recalling that so )

Since is a factor of , the result of polynomial long division of by will be a quotient polynomial with a zero remainder. If this is true, then our original assumption is true, i.e., is a root of because we will have proven that is a factor of

===============================================================
Polynomial Long Division Process
===============================================================
goes into times. is the first term of the quotient.

Multiply times to get and subtract from resulting in . Bring down the to form

goes into times. is the second term of the quotient.

Multiply times to get and subtract from resulting in . Bring down the to form

goes into times. is the third term of the quotient.

Multiply times to get and subtract from resulting in a remainder of zero. The quotient is the sum of the three parts noted above, namely

===============================================================
End of Polynomial Long Division Process
===============================================================

That means that is a factor of , and therefore must also be a factor of , and finally (and its conjugate, )must be a root of .

Since is a quartic (degree 4) polynomial equation, there must be 4 roots. We have found two of them. Since the quotient derived by the polynomial long division process just performed must also be a factor of , the factors of that quotient must also be factors of .

Therefore solving will give the remaining two roots of . I'll leave that one to you. Hint: The quadratic does NOT factor over the integers (or the reals, for that matter), so either complete the square or use the quadratic formula. I completed the square and that worked out rather tidily. Write back and tell me your results.

The following graph of supports the notion that there are no real roots to your given equation because the graph does not intersect the x-axis, so for all real x.

RELATED QUESTIONS
If -2 is a root of z^3-8z^2+9z+58 = 0, then find the other two... (answered by josgarithmetic,MathLover1)

If -2 is a root of {{{ z^3 - 8z^2 + 9z + 58 = 0 }}}, then find the other two roots. (answered by ikleyn)

Consider a polynomial equation of z^4 - z^3 - 5z^2 - z - 6 = 0 . i. Show that z = -i is... (answered by richwmiller)

Check that (2 + i) is a root of (z^4) + (2(z^3)) - (9(z^2)) - 10z + 50 = 0. What are the... (answered by ikleyn,greenestamps)

Check that (2+i) is a root of z^4+2(z^3)-9(z^2)-10z+50=0. Find the remaining three... (answered by MathLover1,MathTherapy)

What is: z^3 --- 35z Times 5 ---- 4z^2 a.)... (answered by lynnlo)

Check that (2+i) is a root of {{{ z^4 + 2z^3 - 9z^2 - 10z + 50 = 0 }}}. Find the... (answered by ikleyn)

Hi, I hope I am submitting this problem under the right section, the chapter I'm... (answered by edjones)

Let tan(A)=1/3 where {{{0 (answered by ikleyn)