if x-√5 is a factor of the cubic polynomial x3-3√5×2+13x-3√5 , then find all the zeroes of the polynomial About the author Faith
Answer: Step-by-step explanation: 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 . Reply
Answer:
Step-by-step explanation:
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
.