Question

Find all zeroes of the polynomial 2x^4−9x³+5x²+3x−1 if two of its zeroes are 2+ √3and 2− √3



the photo uploaded is not the answer of this question but it has the same structure .
please tell me why 2-√3 and 2+√3 is subtrated from x while finding the zeroes
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Answer 1

Answer:

The remaining zeroes of the polynomial 2x⁴ – 9x³ + 5x² + 3x – 1 if two of its zeroes are (2 + √3) and (2 – √3) are – ½ and 1.

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Step-by-step explanation:

According to the question, 2 + √3 and 2 – √3 are two zeroes of the polynomial 2x⁴ – 9x³ + 5x² + 3x – 1.

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We’ll find the zeroes 2 + √3 and 2 – √3 in p(x) form :

☀️ Case ( I ) : For 2 + √3 :

$\sf:\implies~ 2+\sqrt{3}=x$

$\sf:\implies~ x-2-\sqrt{3}$

☀️ Case ( II ) : For 2 – √3 :

$\sf:\implies~ 2-\sqrt{3} = x$

$\sf:\implies~ x-2+\sqrt{3}$

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Now, let’s multiply (x – 2 – √3) and (x – 2 + √3) to get a quadratic polynomial :

( NOTE : The given polynomial is a biquadratic polynomial and has two quadratic polynomials as it’s factors. So, we are multiplying the individual zeroes to get one of the quadratic factor. )

$\sf:\implies~ (x-2-\sqrt{3})~\times~(x-2+\sqrt{3})$

$\sf:\implies~ x~(x-2+\sqrt{3})-2~(x-2+\sqrt{3})-\sqrt{3}~(x-2+\sqrt{3})$

$\sf:\implies~ x^2-2x+\sqrt{3}x-2x+4-2\sqrt{3}-\sqrt{3}x+2\sqrt{3}-3$

$\sf:\implies~ x^2-2x-2x+\sqrt{3}x-\sqrt{3}x-2\sqrt{3}++2\sqrt{3}+4-3$

$\sf:\implies~ x^2-4x+1$

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Dividing the polynomial 2x⁴ – 9x³ + 5x² + 3x – 1 with x² – 4x + 1 to get the remaining quadratic factor :

Refer the attachment!

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Finally, let’s split the middle term of the qoutient we got i.e., 2x² – x – 1 :

$\sf:\implies~ 2x^2-x-1=0$

$\sf:\implies~ 2x^2+x-2x-1=0$

$\sf:\implies~ x~(2x+1)~ -1~(2x+1)=0$

$\sf:\implies~ (2x+1)~(x-1)=0$

$\sf:\implies~ x=1~ (or)~ x=- ^1/_2$

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Therefore, the remaining zeroes of the polynomial 2x⁴ – 9x³ + 5x² + 3x – 1 are – ½ and 1.

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★ Why (2 – √3) and (2 + √3) are subtracted from x while finding the zeroes?

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☀️ Here, (2 – √3) and (2 + √3) are the zeroes of the biquadratic polynomial.

☀️ They have to expressed in the form of a single quadratic equation in order to divide the given biquadratic polynomial and find the other quadratic polynomial.

☀️ So, they are simply not subtracted from x.

☀️ Check out my method in the above one, I’ve done the same but more clearly.

☀️ They are equated to x and from that they get (x – 2 + √3) and (x – 2 – √3).