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Triss [41]
3 years ago
7

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the zeros 4 and 3−

i.
Mathematics
1 answer:
Neko [114]3 years ago
5 0

Answer:

P(x) = x^3 - 10x^2+ 34x-40

Step-by-step explanation:

Given

p =1 -- Leading coefficient

Zeros: 4\ and\ 3  - i

Required

Determine the polynomial

Represent the zeros with a and b

Such that

a = 4

b = 3 - i

The polynomial is:

P(x) = p * (x - a) * (x - b)

P(x) = 1 * (x - 4) * (x - (3 - i))

However, to solve further: We need to symmetrise over 3-i  and its conjugate

P(x) = 1 * (x - 4) * (x - (3 - i))* (x - (3 + i))

P(x) =  (x - 4) * (x - (3 - i))* (x - (3 + i))\\\\

To define a suitable function, the expression becomes

P(x) =  (x - 4) * ((x - 3)^2+1)

P(x) =  (x - 4) * ((x - 3)(x - 3)+1)

P(x) =  (x - 4) * (x^2 - 6x + 9+1)

P(x) =  (x - 4) * (x^2 - 6x +10)

Open bracket

P(x) = x^3 - 6x^2 +10x - 4x^2 + 24x-40

P(x) = x^3 - 6x^2  - 4x^2+10x + 24x-40

P(x) = x^3 - 10x^2+ 34x-40

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