the polynomial f(x)=x4-2x3+3x2-ax+b when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively find the values of - Maths - Polynomials

Question

the polynomial
f(x)=x4-2x3+3x2-ax+b when divided
by (x-1) and (x+1) leaves the remainders 5 and 19 respectively find
the values of a and b hence,find the remainder when f(x) is divided
by (x-2)
find the values of a and b so that the polynomial
(9x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)

Ajanta Trivedi · Accepted Answer

1 the given polynomial is $f (x) = x^{4} - 2 x^{3} + 3 x^{2} - a x + b$ when f(x) is divided by (x-1) remainder is 5 x-1=0 â‡’ x =1 therefore f(1) = 5 $1^{4} - 2 * 1^{3} + 3 * 1^{2} - a * 1 + b = 5 1 - 2 + 3 - a + b = 5 - a + b = 5 - 2 - a + b = 3 (1)$ when f(x) is divided by (x+1) remainder is 19 therefore x+1 = 0 â‡’ x = -1 therefore f( -1) = 19 $(- 1)^{4} - 2 * (- 1)^{3} + 3 * (- 1)^{2} - a * (- 1) + b = 19 1 + 2 + 3 + a + b = 19 a + b = 19 - 6 a + b = 13 (2)$ adding eq(1) and eq(2): $2 b = 3 + 13 2 b = 16 b = \frac{16}{2} = 8$ put b = 8 in eq(1): $- a + 8 = 3 - a = 3 - 8 - a = - 5 \Rightarrow a = 5$ thus the values of a is 5 and b is 8 thus the given polynomial is $f (x) = x^{4} - 2 x^{3} + 3 x^{2} - 5 x + 8$ if f(x) is divided by x-2 , put x-2 = 0 â‡’ x = 2 therefore remainder = f(2) $f (2) = 2^{4} - 2 * 2^{3} + 3 * 2^{2} - 5 * 2 + 8 = 16 - 16 + 12 - 10 + 8 = 10$ 2 let the given polynomial is $f (x) = 9 x^{3} - 10 x^{2} + a x + b$ if f(x) is exactly divisible by (x-1) as well as (x-2) therefore the remainder , when dividing by (x-1) and (x-2) is 0 put x-1 = 0 â‡’ x=1 and x-2 = 0 â‡’ x=2 therefore f(1)=0 and f(2) =0 $f (1) = 0 9 * 1^{3} - 10 * 1^{2} + a * 1 + b = 0 9 - 10 + a + b = 0 a + b - 1 = 0 a + b = 1 (1)$ $f (2) = 0 9 * 2^{3} - 10 * 2^{2} + a * 2 + b = 0 72 - 40 + 2 a + b = 0 32 + 2 a + b = 0 2 a + b = - 32 (2)$ subtracting eq(2) by eq(1): $a - 2 a = 1 + 32 - a = 33 \Rightarrow a = - 33$ put a = -33 in eq(1): $- 33 + b = 1 b = 1 + 33 b = 34$ thus the values of a and b are -33 and 34 respectively hope this helps you cheers!!