Question

Write a polynomial function in factored form of least that the given zeros and a leading coefficient of 1.

1. -6, 3, 5
2. 2, -2, -6i

Write a polynomial function in standard form of least degree that have real coefficients, the given zeros, and a leading coefficient of 1.

3. -1, 1, 7
4. 3, -3, -2i

Answer 1

9514 1404 393

Answer:

1. p(x) = (x+6)(x-3)(x-5)
2. p(x) = (x-2)(x+2)(x^2 +36)
3. p(x) = x^3 -7x^2 -x +7
4. p(x) = x^4 -5x^2 -36

Explanation:

Each root r gives rise to a binomial factor (x-r). Each complex root also has a conjugate root, so roots ±ri give rise to a binomial factor (x^2 +r^2).

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1. p(x) = (x -(-6))(x -3)(x -5)

p(x) = (x+6)(x-3)(x-5)

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2. p(x) = (x -2)(x -(-2))(x^2 +6^2)

p(x) = (x -2)(x +2)(x^2 +36)

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If you want standard form, you can start with factored form, then multiply it out.

3. p(x) = (x -1)(x +1)(x -7) = (x^2 -1)(x -7) = (x^2 -1)x +(x^2 -1)(-7)

p(x) = x^3 -7x^2 -x +7

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4. p(x) = (x -3)(x +3)(x^2 +2^2) = (x^2 -9)(x^2 +4)

p(x) = x^4 -5x^2 -36