The expression 4x3 -px2 + x-q leaves a remainder of 0 and 30 when divided
by (x +1) and (2x - 3) respectively.
Calculate the values of p and q.
Hence factorize the expression completely.
Also find all possible values of x, if 4x2 px2 + x - q = 0.
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Answers
Step-by-step explanation:
Given:-
The expression 4x^3 -px^2 + x-q leaves a remainder of 0 and 30 when divided by (x +1) and (2x - 3) respectively.
To find:-
Calculate the values of p and q.
Hence factorize the expression completely. Also find all possible values of x, if 4x^2-px^2+x-q = 0.
Solution:-
Given expression is 4x^3 -px^2 + x-q
Let P(x) = 4x^3 -px^2 + x-q
If it is divided by (x+1) it leaves the remainder = 0
x+1=0=>x=-1
P(-1) = 0
=>4(-1)^3-p(-1)^2+(-1)-q = 0
=> -4-p-1-q = 0
=> -p-q -5 = 0
=> p+q+5 = 0
=> p = -q -5 -------(1)
and
If it is divided by (2x-3) then it leaves the remainder=30
2x-3 = 0
=>2x = 3
=>x = 3/2
P(3/2)=30
=>4(3/2)^3-p(3/2)^2+(3/2)-q = 30
=>4(27/8)-p(9/4)+(3/2)-q = 30
=> (27/2)-(9p/4)+(3/2)-q = 30
=> (54-9p+6-4q)/4 = 30
=> (60-9p-4q)/4 = 30
=> 60-9p-4q = 30×4
=>60-9p-4q = 120
=> -9p-4q = 120-60
=> -9p-4q = 60
=> 9p+4q = -60
from(1)
=> 9(-q-5)+4q = -60
=> -9q-45+4q = -60
=>-5q-45 = -60
=>-5q = -60+45
= > -5q = -15
=>q = -15/-5
=> q = 3
from (1)
p = -3-5
=> p =-8
The values of p and q are -8 and 3
respectively.
Then the expression becomes
4x^3-(-8)x^2+x-(3)
=>4x^3+8x^2+x-3
=> 4x^3+4x^2+4x^2+4x-3x-3
=> 4x^2(x+1)+4x(x+1)-3(x+1)
=> (x+1)(4x^2+4x-3)
=>(x+1)(4x^2+4x-3)
=> (x+1)(4x^2-2x+6x-3)
=> (x+1)[2x(2x-1)+3(2x-1)]
=> (x+1)(2x-1)(2x+3)
Now 4x^3+8x^2+x-3 = 0
=>(x+1)(2x-1)(2x+3) =0
=>x+1 = 0 or 2x-1 = 0 or 2x+3 = 0
=> x = -1 or x= 1/2 or x = -3/2
Answer:-
The value of p = -8
The value of q = 3
Factorization of the given expression is
(x+1)(2x-1)(2x+3)
The possible values of x are -1 , 1/2 and -3/2