a polynomial f(x) degree is greater than 3 leaves the remainder 2,1, -1 when divided by x-1,x+2 and x+1 respectively what will be the remainder when it is divided by x-1 ×x+2×x-1
Answers
Answer:
x² + (3/2) x
[ I have taken the liberty of "correcting" the question, which I'm sure was meant to say "divded by (x-1)(x+2)(x+1)" ; i.e. the last factor is (x+1). ]
Step-by-step explanation:
Since the degree of (x-1)(x+2)(x+1) is 3, the remainder has degree at most 2, so it has the form ax²+bx+c. We just need to work out a, b and c.
Doing the division, since ax²+bx+c is the remainder, we get
f(x) = ( x - 1 ) ( x + 2 ) ( x + 1 ) g(x) + ax² + bx + c
where g(x) is some polynomial we don't care about here.
By the Remainder Theorem, f(a) is the remainder when we divide by (x-a), so putting in x = 1, x = -2 and x = -1 one at a time, we get:
2 = f(1) = a + b + c (i)
1 = f(-2) = 4a - 2b + c (ii)
-1 = f(-1) = a - b + c (iii)
Subtracting (iii) from (i) gives 2b = 3, so b = 3/2.
Putting this into (ii) gives
1 = 4a - 3 + c, so 4 = 4a + c (iv)
Adding (i) and (iii) gives
1 = a + c (v)
Subtracting (v) from (iv) gives
3 = 3a, so a = 1.
Finally, putting this into (v) gives c = 0.
Therefore the remainder is
ax² + bx + c = x² + (3/2) x
Hope this helps.