Find a polynomial function of degree n with only the following real zeros for each set of zeros. 57. -5,4;n=4 61. 0,3,-2;n=5

Question

Find a polynomial function of degree
`n` with only the following real zeros for each set of zeros.

57.
`-5,4;n=4`
61.
`0,3,-2;n=5`

Answer 1

9514 1404 393

Answer:

57. p(x) = x^4 +x^3 -19x^2 +x -20

61. p(x) = x^5 -x^4 -5x^3 -x^2 -6x

Explanation:

For real zero "q", the polynomial will have a linear factor (x -q). For complex zeros, the polynomial will need a quadratic factor that has no real zeros. A simple such factor is (x^2 +1).

57. p(x) = (x -(-5))(x -4)(x^2 +1)

p(x) = x^4 +x^3 -19x^2 +x -20

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61. p(x) = (x -0)(x -3)(x -(-2))(x^2 +1)

p(x) = x^5 -x^4 -5x^3 -x^2 -6x