Find the values of a and b so that x⁴ + x³ + 8x² + ax + b is divisible by x² + 1 .
Answers
Answer:
The expression is is divisible by
TO FIND :
The values of a and b by solving the given polynomials by Long Division method
SOLUTION :
Now divide the given polynomial
x^4+x^3+8x^2+ax-bx
4
+x
3
+8x
2
+ax−b is divisible by x^2+1x
2
+1
x^2+1x
2
+1 can be written as x^2+0x+1x
2
+0x+1
x^2+x+7x
2
+x+7
______________________
x^2+0x+1x
2
+0x+1 ) x^4+x^3+8x^2+ax-bx
4
+x
3
+8x
2
+ax−b
x^4+0x^3+x^2x
4
+0x
3
+x
2
___(-)___(-)__(-)_____________
x^3+7x^2+axx
3
+7x
2
+ax
x^3+0x^2+xx
3
+0x
2
+x
__(-)__(-)__(-)_____
7x^2+(a-1)x-b7x
2
+(a−1)x−b
7x^2+0x+77x
2
+0x+7
_(-)___(-)__(-)___
(a-1)x-b-7
_________
The quotient is and remainder is (a-1)x-b-7
Since the the polynomial is completely divided by so that the remainder is zero
∴ (a-1)x-b-7=0
(a-1)x-b-7=0 can be written as
(a-1)x+(-b-7)=0x+0
Now equating the coefficients of x and constant we get
a-1=0 and -b-7=0
∴ a=1 and b=-7
∴ The values of a and b is 1 and -7 respectively.
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If x⁴ + x³ + 8x² + ax + b is divisible by x² + 1 ,then the remainder should be zero.
On dividing, we get:
x² + 1 ) x⁴ + x³ + 8x² + ax + b ( x² + x + 7
x⁴ + x²
- -
x³ + 7x² + ax + b
x³ + x
- -
7x² + x ( a - 1 ) + b
7x² + 7
- -
x ( a - 1 ) + ( b - 7 )
Therefore,
★ Quotient = x² + x + 7
★ Remainder = x ( a - 1 ) ( b - 7 )
Now,
Remainder = 0
⇒ a - 1 = 0 and b - 7 = 0
⇒ a = 1 and b = 7