if f(x)=x^4-2x^3+3x^2-ax+b is divided by x-1 and x+1 the remainder are 5 and 19 respectively, then find the reminder when(x) is divided by(x-2)
Answers
Answer:
a = 5
b = 8
Remainder = 10
Step-by-step explanation:
There is theorem known as “Polynomial Remainder Theorem” or “ Bezout’s Theorem”. It is Stated as -
A Polynomial f(x) if divided by a linear polynomial (x-a) leaves remainder which equals f(a).
So , getting back to our question -
f(x) = x^4 - 2x^3 + 3x^2 - ax + b
So , when it is divided by (x - 1) it’ll leave a remainder = f(1) = 5 (Given).
f(1) = 1^4 - 2×1^3 + 3×1^2 - a×1 + b = 5
=> 1 - 2 + 3 - a + b = 5
=> a - b = (-3) …. Eqn(1)
Now , Similarly -
f(-1) = (-1)^4 - 2×(-1)^3 + 3×(-1)^2 - a×(-1) + b = 19
=> 1 + 2 + 3 + a + b = 19
=> a + b = 13 …. Eqn(2)
Now , adding equations (1) and (2) , We’ll get -
(a+b) + (a-b) = (-3) + 13
=> 2a = 10 => a = 5
So , (a +b) = 13 implies b = 8
Hence , Values of a and b are 5 and 8 respectively.
Hence, our
p(x) = x^4 - 2x^3 + 3x^2 - 5x + 8
Now,
Dividend p(x) = x^4 - 2x^3 + 3x^2 - 5x + 8
Divisor g(x) = x - 2
By using remainder theorem, we get,
x - 2 = 0
=> x = 2
By applying the values of x in p(x), we get,
(2)^4 - 2×(2)^3 + 3×(2)^2 - (5×2) + 8
=> 16 - 16 + 12 - 10 + 8
=> 2 + 8
=> 10
Hence, the remainder r(x), when x^4 - 2x^3 + 3x^2 - 5x + 8 is divided by (x-2) is 10
Answer:
20
Step-by-step explanation:
Explanation attached
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