In the North-East of India Flood affected a village. Mr. Sharma from Mumbai decided to help them with food items, clothes, medicines etc. so, he collected some amount from different persons, which is represented by 8x^4 + 14x^3 - 2x^2 + ax - b. If 4x^2 + 3x - 2 number of person were helped by this service and left over amount of 14x - 10 was donated in Prime Minister Relief Fund. Find the value of a and b. What values are possessed by Mr. Sharma?
Answers
Answer:
2xsquare
Answer:
a=1 and b=7
Step-by-step explanation:
Given,
Total amount of money collected = (x^{4}+x^{3}+8x^{2}+ax+b)(x
4
+x
3
+8x
2
+ax+b)
No of person = (x^{2}+1)(x
2
+1)
As the total money collected given by (x^{2}+1)(x
2
+1) no of person.
so (x^{4}+x^{3}+8x^{2}+ax+b)(x
4
+x
3
+8x
2
+ax+b) is divisible by (x^{2}+1)(x
2
+1)
Let
p(x)=(x^{4}+x^{3}+8x^{2}+ax+b)p(x)=(x
4
+x
3
+8x
2
+ax+b)
g(x)=(x^{2}+1)g(x)=(x
2
+1)
Since g(x)g(x) divides p(x)p(x)
therefore the quotient q(x)q(x) is a polynomial of degree 2
Consider,
q(x)=a_{1}x^{2}+b_{1}x+c_{1}q(x)=a
1
x
2
+b
1
x+c
1
So we can write the equation as
\begin{gathered}p(x)=g(x)\times{q}(x)\\\\\Rightarrow(x^{4}+x^{3}+8x^{2}+ax+b)=(x^{2}+1)(a_{1}x^{2}+b_{1}x+c_{1})\\\\\Rightarrow(x^{4}+x^{3}+8x^{2}+ax+b)=(a_{1}x^{4}+b_{1}x^{3}+c_{1}x^{2}+a_{1}x^{2}+b_{1}x+c_{1})\\\\\Rightarrow(x^{4}+x^{3}+8x^{2}+ax+b)=a_{1}x^{4}+b_{1}x^{3}+(c_{1}+a_{1})x^{2}+b_{1}x+c_{1}\end{gathered}
p(x)=g(x)×q(x)
⇒(x
4
+x
3
+8x
2
+ax+b)=(x
2
+1)(a
1
x
2
+b
1
x+c
1
)
⇒(x
4
+x
3
+8x
2
+ax+b)=(a
1
x
4
+b
1
x
3
+c
1
x
2
+a
1
x
2
+b
1
x+c
1
)
⇒(x
4
+x
3
+8x
2
+ax+b)=a
1
x
4
+b
1
x
3
+(c
1
+a
1
)x
2
+b
1
x+c
1
By comparing co-efficient of x^{4}x
4
on both side we get
a_{1}=1a
1
=1
By comparing co-efficient of x^{3}x
3
on both side we get
b_{1}=1b
1
=1
By comparing co-efficient of x^{2}x
2
on both side we get
\begin{gathered}a_{1}+c_{1}=8\\\\\Rightarrow{c_{1}}=8-a_{1}\\\\\Rightarrow{c_{1}}=8-1=7\end{gathered}
a
1
+c
1
=8
⇒c
1
=8−a
1
⇒c
1
=8−1=7
By comparing co-efficient of xx on both side we get
\begin{gathered}b_{1}=a\\\\\Rightarrow1=a\\\\\Rightarrow{a}=1\end{gathered}
b
1
=a
⇒1=a
⇒a=1
By comparing constant term of both side we get
\begin{gathered}b=c_{1}\\\\\Rightarrow{b}=7\end{gathered}
b=c
1
⇒b=7
plz mark me brainlist