FP1 Complex numbers help

Bit stuck on a question where I've been given the roots of a cubic equation (2, 1+5i and 1-5i) and have to find coefficients of the x^2 and x values in the original equation... How do I go about doing that? =s
Thanks in advance.

Reply 1

TeeEm

19

Original post by Jakkson

Bit stuck on a question where I've been given the roots of a cubic equation (2, 1+5i and 1-5i) and have to find coefficients of the x^2 and x values in the original equation... How do I go about doing that? =s
Thanks in advance.

[x-2][x-(1+5i)][x-(1-5i)]

Reply 2

davros

16

Original post by Jakkson

Bit stuck on a question where I've been given the roots of a cubic equation (2, 1+5i and 1-5i) and have to find coefficients of the x^2 and x values in the original equation... How do I go about doing that? =s
Thanks in advance.

You can multiply out explicitly as TeeEm suggests, or use general rules about sums and products of roots of an equation e.g. if (x - a)(x -b)(x- c) is your factorized cubic with zeros at a, b and c, then the coeff of x^2 in the expansion is -(a + b + c) etc.

Reply 3

WillWalker23

7

Although it is the next chapter, using roots of polynomials is useful here. It makes less room for error compared with expanding bracket.

-b/a = alpha + beta + gamma = 2 + 1 + 5i + 1 - 5i => -b/a = 4. Therefore b=-4

c/a =alpha.beta + alpha.gamma + beta.gamma = 2(1+5i) + 2(1-5i) + (1+5i)(1-5i) = 2 + 10i + 2 + 10i + 26 =30. Therefore c=30

-d/a = alpha.beta.gamma = 2(1+5i)(1-5i) = 2*26 = 52. Therefore d=-52

So a must be 1. Therefore the equation (ax^3 + bx^2 + cx +d) is: x^3 - 4x^2 + 30x - 52

FP1 Complex numbers help

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