If x4 + 3x2 - 7 is divided by 3x + 5 then the possible degrees of quotient and remainder are Select one: a. 3,1 b. 4,1 c. 4,0 d. 3,0
Answers
Answer:
On this division , what is the possible degrees of quotient and remainder ? Hence , option (A) is correct answer i.e. degree of quotient is 3 and degree of remainder is 1
Answer:
Given:
p(x)=x
4
−3x
2
+4x+5
g(x)=x
2
+1−x
Degree of q(x)= Degree of p(x)−Degree of g(x)
=4−2=2
Degree of r(x)< Degree of g(x)
Let degree of r(x)=1
Let q(x)=ax
2
+bx+c and r(x)=px+q
By division algorithm
p(x)=q(x)×g(x)+r(x)
x
4
−3x
2
+4x+5=(ax
2
+bx+c)(x
2
+1−x)+(px+q)
Comparing the coefficent of x
4
.
a=1.
Comparing the coefficent of x
3
.
−a+b=0⇒b=a⇒b=1
Comparing the coefficent of x
2
a−b+c=−3
⇒1−1+c=−3⇒c=−3
Comparing the coefficent of x.
b−c+p=4
⇒1+3+p=4
⇒p=0
Comparing the coefficent of constant term.
c+q=5
−3+q=5
q=8.
So, quotient q(x)=ax
2
+bx+c
=x
2
+x−3
and remainder r(x)=px+q
=8