If $(3 - \sqrt{2})$ $(3-sqrt2)$ and $(2 + \sqrt{2})$ $(2+sqrt2)$ are two of the roots of a fourth-degree polynomial with integer coefficients, what is the product of the other two roots?

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If $(3 - \sqrt{2})$ $(3-sqrt2)$ and $(2 + \sqrt{2})$ $(2+sqrt2)$ are two of the roots of a fourth-degree polynomial with integer coefficients, what is the product of the other two roots?

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Answer 1 · 2016-12-25T14:30:30

$4 - \sqrt{2}$ $4-sqrt(2)$

Explanation:

The two other roots will be the radical conjugates of the given ones, namely:

$(3 + \sqrt{2})$ $(3+sqrt(2))$ and $(2 - \sqrt{2})$ $(2-sqrt(2))$

So (using FOIL) their product is:

$(3+sqrt(2))(2-sqrt(2)) = overbrace(3*2)^"First"+overbrace(3(-sqrt(2)))^"Outside"+overbrace((sqrt(2))2)^"Inside"+overbrace((sqrt(2))(-sqrt(2)))^"Last"$

$color(white)((3+sqrt(2))(2-sqrt(2))) = 6-3sqrt(2)+2sqrt(2)-2$

$color(white)((3+sqrt(2))(2-sqrt(2))) = 4-sqrt(2)$

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If $(3 - \sqrt{2})$ $(3-sqrt2)$ and $(2 + \sqrt{2})$ $(2+sqrt2)$ are two of the roots of a fourth-degree polynomial with integer coefficients, what is the product of the other two roots?

1 Answer

Explanation:

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If (3-sqrt2)(3−√2)(3-sqrt2) and (2+sqrt2)(2+√2)(2+sqrt2) are two of the roots of a fourth-degree polynomial with integer coefficients, what is the product of the other two roots?

1 Answer

Explanation:

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Impact of this question

If $(3 - \sqrt{2})$ $(3-sqrt2)$ and $(2 + \sqrt{2})$ $(2+sqrt2)$ are two of the roots of a fourth-degree polynomial with integer coefficients, what is the product of the other two roots?