if x-√5 is a factor of the cubic polynomial x3-3√5x2+13x-3√5 , then find all the zeroes of the polynomial
Answers
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
.