Question

Find zeros of polynomial

$\\ \sqrt[5]{5} {x}^{2} + 30x + \sqrt[8]{5}$
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Answer 1

Required Answer:-

$\\\longrightarrow \tt \: \sqrt[5]{5} {x}^{2} + 30x + \sqrt[8]{5}$

➡️By splitting the middle term,

$\\ \longrightarrow \tt \: \sqrt[5]{5} {x}^{2} + 20x + 10x + \sqrt[8]{5}$

$\\ \longrightarrow \tt \: \sqrt[5]{5} {x}^{2} + (5 \times 4)x + (5 \times 2)x + \sqrt[8]{5}$

$\\ \longrightarrow \tt \: \sqrt[5]{5} {x}^{2} + (\sqrt{5} \times \sqrt{5} \times 4)x + (5 \times 2)x + \sqrt[8]{5}$

$\\ \longrightarrow \tt \: \sqrt[5]{x} \times (5x + \sqrt[4]{5}) + 2(5x + \sqrt[4]{5})$

$\\ \longrightarrow \tt \: (5x + \sqrt[4]{5}) \: ( \sqrt{5}x + 2)$

$\\ \longrightarrow \tt \: (5x + \sqrt[4]{5}) \implies(1)$

➡️From (1)

$\\ \longrightarrow \tt \: 5x = - \sqrt[4]{5}$

$\\ \longrightarrow \tt \: x = \frac{ - \sqrt[4]{5} }{5}$

$\longrightarrow \tt \: ( \sqrt{5}x + 2) \implies \: (2)$

➡️From (2)

$\\ \longrightarrow \tt \: \sqrt{5}x + 2 = 0$

$\\ \longrightarrow \tt \: \sqrt{5}x = - 2$

$\\ \longrightarrow \tt \: x = \frac{ - 2}{ \sqrt{5} }$

➡️The two Zeroes of the polynomial is :-

$\\ \longrightarrow \tt \: \frac{ - \sqrt[4]{5} }{5} , \frac{ - 2}{ \sqrt{5} }$

Answer 2

Answer:

Zeroes of the polynomial are$\frac{ - 2\sqrt{5} }{5}, \frac{ - 4\sqrt{5} }{5}$

Step-by-step explanation:

Given equation,

5 root 5 x square + 30x + 8 root 5

This can be written as,

5√5x²+ 30 x + 8√5

We will factorise the given equation by splitting the middle term method

5√5 x² + 20 x + 10 x + 8√5

10 x can be written as ( 5 * 2) x

5√5 x² + 20 x + (5*2) x + 8√5

Also, 5 can be written as √5 * √5

So now the Equation becomes :

= 5√5 x² + 20 x+ (√5 * √5 * 2)x+ 8√5 = 5x ( √5 x + 4 ) + √5 * 2 ( √5 x+ 4)

= 5x ( √5 x + 4) + 25 ( √5 x + 4) = ( 5x + 2√5 ) (√5 x + 4 )

Zeroes are

5x + 2√5 =0

5x = -2√5

$\frac{ - 2 \sqrt{5} }{5}$

Also,

√5 x + 4 = 0

√5 x = - 4

$x = \frac{ - 4}{ \sqrt{5} }$

Multiplying and dividing by √5

$x = \frac{ - 4}{ \sqrt{5} } \times \frac{ \sqrt{5} }{ \sqrt{5} }$

$x = \frac{ - 4 \sqrt{5} }{5}$

Hence,

Zeroes of the polynomial are -2v5, -15