Question

Given that x – √5 is a factor of the cubic polynomial x3 – 3√5x2 +13x – 3√5, find all the zeroes of the polynomial

Answer 1

Answer:

Hence, the zeros of the given expression are  

5​+  2​  ,  5​  −  2​  ,  5

 

Correct option is

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 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

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