Question

find the zero polynomial
$x - \sqrt{5}$
factor of
${x}^{3} - 3 \sqrt{ {5x}^{2} } + 5x + 15 \sqrt{5}$
​

Answer 1

Answer:

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Step-by-step explanation:

Answer

A

5

,

5

+

2

,

5

−

2

If (x−

5

) is a factor, then we can write:

x

3

–3

5

x

2

+13x–3

5

=(x–

5

)(x

2

+bx+3)

To determine the coefficient b, let's expand the product:

(x–

5

)(x

2

+bx+3)=x

3

+bx

2

+3x–(

5

)x

2

–(

5

)bx–3

5

(x–

5

)(x

2

+bx+3)=x

3

+(b–

5

)x

2

+(3–b

5

)x–3

5

Comparing the right hand side to the original expression, we obtain

b–

5

=−3

5

⇒b=−2

5

, or, with the same result:

3–b

5

=13

⇒b

5

=−10

⇒b=−10/

5

=−2

5

⇒b=−2

5

Therefore,

x

3

–3

5

x

2

+13x–3

5

=(x–

5

)(x

2

–2

5

x+3)

x

3

–3

5

x

2

+13x–3

5

=0

(x–

5

)=0,(x

2

–2

5

x+3)=0

x–

5

=0⇒x=

5

x

2

–2

5

x+3=0⇒x=

5

±

2

Hence, the zeros of the given expression are

5

+

2

,

5

−

2

,

5

.