Question

Find all zeros of the polynomial f(x) = 2x² - 2x³-7x² + 3x + 6, if its two zeros are ✓-3/2 and ✓3/2​

Answer 1

Given: We are given with two zeroes of polynomial are √-3/2 and √3/2 and f(x)=2x^2-2x^3-7x^2+3x+6.

To Find: Find all zeroes of the polynomial.

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$\: \sf \: f(x) = 2 {x}^{4} - 2 {x}^{3} - 7 {x}^{2} + 3x + 6 \\ \\ \sf - \sqrt{ \frac{3}{2} } and \sqrt{ \frac{3}{2} } are \: zeroes \: of \: f(x).$

Now find the factor of the zeroes which are given:

$\sf \: factor = (x + \sqrt{ \frac{3}{2} } )(x - \sqrt{ \frac{3}{2} } )$

Now multiply both terms

$\: \sf \: = {x}^{2} - \frac{3}{2}$

Now take LCM

$\: \sf \: = \frac{2 {x}^{2} - 3 }{2}$

Now take common numbers

$\: \sf \: = \frac{1}{2} (2 {x}^{2} - 3)$

After dividing the term 2x^4-2x^3-7x^2+3x+6 from 2x^2-3 we got x^2-x-2.

Now put on f(x) value

$\: \sf \: f(x) = 2 {x}^{2} - 3( {x}^{2} - x - 2) \\ \\ \: \sf \: f(x) = 2 {x}^{2} - 3( {x}^{2} - 2x + x - 2) \\ \\ \: \sf \: f(x) = 2 {x}^{2} - 3(x(x - 2) + (x - 2)) \\ \\ \: \sf \: f(x) = 2 {x}^{2} - 3(x - 2)(x + 1)$

$\: \sf \therefore \: x - 2 \\ \sf \: \: x = 2 \\ \\ \: \sf \therefore \: x + 1 \\ \sf \: x = - 1$

Hence,zeroes of the polynomial are 2 and -1.

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Answer 2

$\huge\mathfrak\red{Question}$

Find all zeros of the polynomial f(x) = are 2x² - 2x³ - 7x² + 3x + 6, if it's two zeros are √-3/2 and √3/2.

$\huge\mathfrak\red{Answer}$

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$\begin{gathered} \: \sf \: f(x) = 2 {x}^{4} - 2 {x}^{3} - 7 {x}^{2} + 3x + 6 \\ \\ \sf - \sqrt{ \frac{3}{2} } and \sqrt{ \frac{3}{2} } are \: zeroes \: of \: f(x).\end{gathered}$

Now find the factor of the zeroes which are given:

$\sf \: factor = (x + \sqrt{ \frac{3}{2} } )(x - \sqrt{ \frac{3}{2} } )factor$

Now multiply both terms

$\: \sf \: = {x}^{2} - \frac{3}{2}=x$

Now take LCM

$\: \sf \: = \frac{2 {x}^{2} - 3 }{2}=$

Now take common numbers

$\: \sf \: = \frac{1}{2} (2 {x}^{2} - 3)$

After dividing the term 2x⁴ - 2x³- 7x² + 3x + 6 from 2x² - 3 we got x² - x - 2.

Now put on f(x) value

$\begin{gathered} \: \sf \: f(x) = 2 {x}^{2} - 3( {x}^{2} - x - 2) \\ \\ \: \sf \: f(x) = 2 {x}^{2} - 3( {x}^{2} - 2x + x - 2) \\ \\ \: \sf \: f(x) = 2 {x}^{2} - 3(x(x - 2) + (x - 2)) \\ \\ \: \sf \: f(x) = 2 {x}^{2} - 3(x - 2)(x + 1)\end{gathered}$

$\begin{gathered} \: \sf \: x - 2 \\ \sf \: \: x = 2 \\ \\ \: \sf \: x + 1 \\ \sf \: x = - 1\end{gathered}$

Hence,zeroes of the polynomial are 2 and -1.

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