If you know that $3 + \sqrt{11}$ $3 + sqrt(11)$ is a root of a polynomial function, then the name given to $3 - \sqrt{11}$ $3 - sqrt(11)$ , another root of the same function , is a __ conjugate. ?

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If you know that $3 + \sqrt{11}$ $3 + sqrt(11)$ is a root of a polynomial function, then the name given to $3 - \sqrt{11}$ $3 - sqrt(11)$ , another root of the same function , is a __ conjugate. ?

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Answer 1 · 2017-12-17T19:04:32

radical conjugate

Explanation:

A conjugate is an object which when combined with the original object makes some kind of whole.

Note that the difference of squares identity tells us that:

$a^{2} - b^{2} = (a - b) (a + b)$ $a^2-b^2 = (a-b)(a+b)$

So if $a$ $a$ and $b$ $b$ are terms consisting of square roots (possibly including $i$ $i$ which is a square root of $- 1$ $-1$ ), then we can simplify $(a + b)$ $(a+b)$ by multiplying by the conjugate $(a - b)$ $(a-b)$ (or vice versa).

In the case of square roots, this is a radical conjugate.

For example, we find:

$(x - (3 - \sqrt{11})) (x - (3 + \sqrt{11})) = ((x - 3) - \sqrt{11}) ((x - 3) + \sqrt{11})$ $(x-(3-sqrt(11)))(x-(3+sqrt(11))) = ((x-3)-sqrt(11))((x-3)+sqrt(11))$

$(x - (3 - \sqrt{11})) (x - (3 + \sqrt{11})) = {(x - 3)}^{2} - {(\sqrt{11})}^{2}$ $color(white)((x-(3-sqrt(11)))(x-(3+sqrt(11)))) = (x-3)^2-(sqrt(11))^2$

$(x - (3 - \sqrt{11})) (x - (3 + \sqrt{11})) = x^{2} - 6 x + 9 - 11$ $color(white)((x-(3-sqrt(11)))(x-(3+sqrt(11)))) = x^2-6x+9-11$

$(x - (3 - \sqrt{11})) (x - (3 + \sqrt{11})) = x^{2} - 6 x - 2$ $color(white)((x-(3-sqrt(11)))(x-(3+sqrt(11)))) = x^2-6x-2$

Note that the resulting quadratic polynomial has only rational coefficients - we have successfully eliminated the irrational $\sqrt{11}$ $sqrt(11)$ by using the radical conjugate.

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If you know that $3 + \sqrt{11}$ $3 + sqrt(11)$ is a root of a polynomial function, then the name given to $3 - \sqrt{11}$ $3 - sqrt(11)$ , another root of the same function , is a __ conjugate. ?

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Explanation:

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If you know that 3 + sqrt(11)3+√113 + sqrt(11) is a root of a polynomial function, then the name given to 3 - sqrt(11)3−√113 - sqrt(11) , another root of the same function , is a __ conjugate. ?

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If you know that $3 + \sqrt{11}$ $3 + sqrt(11)$ is a root of a polynomial function, then the name given to $3 - \sqrt{11}$ $3 - sqrt(11)$ , another root of the same function , is a __ conjugate. ?