Advertisements
Advertisements
प्रश्न
When a polynomial f(x) is divided by (x – 1), the remainder is 5 and when it is,, divided by (x – 2), the remainder is 7. Find – the remainder when it is divided by (x – 1) (x – 2).
Advertisements
उत्तर
When f(x) is divided by (x – 1),
Remainder = 5
Let x – 1 = 0
⇒ x = 1
∴ f(1) = 5
When divided by (x – 2),
Remainder = 7
Let x – 2 = 0
⇒ x = 2
∴ f(2) = 7
Let f(x) = (x – 1)(x – 2)q(x) + ax + b
Where q(x) is the quotient and ax + b is remainder
Putting x = 1, we get:
f(1) = (1 – 1)(1 – 2)q(1) + a x 1 + b
= 0 + a + b
= a + b
and x = 2, then
f(2) = (2 – 1)(2 – 2)q(2) + a x 2 + b
= 0 + 2a + b
= 2a + b
∴ a + b = 5 ....(i)
2a + b = 7 ....(ii)
Subtracting, we get
–a = – 2
⇒ a = 2
Substituting the value of a in (5)
2 + b = 5
⇒ b = 5 – 2 = 3
∴ a = 2, b = 3
∴ Remainder = ax + b
= 2x + 3.
APPEARS IN
संबंधित प्रश्न
Find the remainder when x3 + 3x2 + 3x + 1 is divided by 5 + 2x.
Find the remainder when x4 – 3x2 + 2x + 1 is divided by x – 1.
If x3 + ax2 + bx + 6 has x – 2 as a factor and leaves a remainder 3 when divided by x – 3, find the values of a and b.
What number should be subtracted from x2 + x + 1 so that the resulting polynomial is exactly divisible by (x-2) ?
If p(x) = 4x3 - 3x2 + 2x - 4 find the remainderwhen p(x) is divided by:
x - 4
Using remainder theorem, find the remainder on dividing f(x) by (x + 3) where f(x) = 2x2 – 5x + 1
Find the remainder (without division) on dividing f(x) by (2x + 1) where f(x) = 3x3 – 7x2 + 4x + 11
Find ‘a’ if the two polynomials ax3 + 3x2 – 9 and 2x3 + 4x + a, leaves the same remainder when divided by x + 3.
If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + a leave the same remainder when divided by z – 3, find the value of a.
4x2 – kx + 5 leaves a remainder 2 when divided by x – 1. The value of k is ______.
