If the polynomial 4 − 63 − 262 + 138 − 35 divided by another polynomial
2− 4x+ k. The reminder comes out to be zero find k.?
Answers
Answer:
The remaining zeores are -5 and 7
Step-by-step explanation:
The given polynomial is
P(x)=x^4-6x^3-26x^2+138x-35P(x)=x
4
−6x
3
−26x
2
+138x−35
It is given that 2+\sqrt{3}2+
3
and 2-\sqrt{3}2−
3
are two zeros. It means (x-2-\sqrt{3})(x−2−
3
) and (x-2+\sqrt{3})(x−2+
3
) are factors of the given polynomial.
(x-2-\sqrt{3})(x-2+\sqrt{3})=(x-2)^2-3=x^2-4x+4-3=x^2-4x+1(x−2−
3
)(x−2+
3
)=(x−2)
2
−3=x
2
−4x+4−3=x
2
−4x+1
Divide the given polynomial by (x^2-4x+1).
\frac{x^4-6x^3-26x^2+138x-35}{x^2-4x+1}=x^2-2x-35
x
2
−4x+1
x
4
−6x
3
−26x
2
+138x−35
=x
2
−2x−35
It means the remaining factor of P(x) is x^2-2x-35x
2
−2x−35 .
x^2-2x-35=0x
2
−2x−35=0
x^2-7x+5x-35=0x
2
−7x+5x−35=0
x(x-7)+5(x-7)=0x(x−7)+5(x−7)=0
(x+5)(x-7)=0(x+5)(x−7)=0
Equate each factor equal to 0.
x=-5,x=7x=−5,x=7