How do you construct polynomial equations with the given roots?

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How do you construct polynomial equations with the given roots?

1. $2$ $2$ , $4$ $4$ and $- 7$ $-7$ .
2. $5$ $5$ and $\sqrt{3}$ $sqrt(3)$ .

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Answer 1 · 2017-08-11T18:25:06

1. $x^{3} + x^{2} - 34 x + 56 = 0$ $x^3+x^2-34x+56 = 0$

2. $x^{3} - 5 x^{2} - 3 x + 15 = 0$ $x^3-5x^2-3x+15 = 0$

Explanation:

Note that if a polynomial in $x$ $x$ has a zero $a$ $a$ then it has a factor $(x - a)$ $(x-a)$ and vice versa.

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For question 1 we can construct a polynomial:

$f (x) = (x - 2) (x - 4) (x + 7) = x^{3} + x^{2} - 34 x + 56$ $f(x) = (x-2)(x-4)(x+7) = x^3+x^2-34x+56$

Any polynomial with these zeros will be a multiple (scalar or polynomial) of this $f (x)$ $f(x)$ .

So the polynomial equation:

$x^{3} + x^{2} - 34 x + 56 = 0$ $x^3+x^2-34x+56 = 0$

has roots $2$ $2$ , $4$ $4$ and $- 7$ $-7$ .

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For question 2 we can multiply out $(x - 5) (x - \sqrt{3})$ $(x-5)(x-sqrt(3))$ but this will result in a polynomial with irrational coefficients:

$(x - 5) (x - \sqrt{3}) = x^{2} - (5 + \sqrt{3}) x + 5 \sqrt{3}$ $(x-5)(x-sqrt(3)) = x^2-(5+sqrt(3))x+5sqrt(3)$

If - as is probably the case - we want a polynomial with integer coefficients, then we also need the rational conjugate $- \sqrt{3}$ $-sqrt(3)$ to be a zero and $(x + \sqrt{3})$ $(x+sqrt(3))$ a factor.

Then we can define:

$g (x) = (x - 5) (x - \sqrt{3}) (x + \sqrt{3}) = (x - 5) (x^{2} - 3) = x^{3} - 5 x^{2} - 3 x + 15$ $g(x) = (x-5)(x-sqrt(3))(x+sqrt(3)) = (x-5)(x^2-3) = x^3-5x^2-3x+15$

Any polynomial with these zeros will be a multiple (scalar or polynomial) of this $g (x)$ $g(x)$ .

So the polynomial equation:

$x^{3} - 5 x^{2} - 3 x + 15 = 0$ $x^3-5x^2-3x+15 = 0$

has roots $5$ $5$ , $\sqrt{3}$ $sqrt(3)$ and $- \sqrt{3}$ $-sqrt(3)$ .

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How do you construct polynomial equations with the given roots?

1. $2$ $2$ , $4$ $4$ and $- 7$ $-7$ .
2. $5$ $5$ and $\sqrt{3}$ $sqrt(3)$ .

1 Answer

Explanation:

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Humanities

... and beyond

How do you construct polynomial equations with the given roots?

1. 22, 44 and −7-7. 2. 55 and √3sqrt(3).

1 Answer

Explanation:

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Impact of this question

1. $2$ $2$ , $4$ $4$ and $- 7$ $-7$ .
2. $5$ $5$ and $\sqrt{3}$ $sqrt(3)$ .