Polynomials

2014-11-01 12:32 am

The remainder when a polynomial f(x) is divided by (x-2)(x+3) is ax+b.
When f(x) is divided by x-2, the remainder is 5.
x+3 is a factor of f(x).
Find values of a & b.

With full steps please!

更新1:

One more please! When a polynomial f(x) is divided by x-5, the remainder is 9. When f(x) is divided by x+2, the remainder is -5. Find the remainder when f(x) is divided by (x-5)(x+2).

回答 (1)

邊位都好

2014-11-01 1:51 am

✔ 最佳答案

1. Let f(x) = (x-2)(x+3)q(x) + ax+b
as f(2) = 5, so
5 = (2-2)(2+3)q(2) + 2a+b
==> 2a + b = 5 ⋯⋯ (i)
and f(-3) = 0, so
0 = (-3-2)(-3+3)f(-3) + (-3)a+b
==> 3a - b = 0 ⋯⋯ (ii)
solving (i), (ii), we get,
a = 1, b = 3

2. Let the remainder be (ax+b) when f(x) is divided by (x-5)(x+2),
so, f(x) = (x-5)(x+2)q(x) + ax+b
as f(5)=9, so
9 = (5-5)(5+2)q(5) + 5a+b
==> 5a + b = 9 ⋯⋯ (iii)
and f(-2) = -5, so
-5 = (-2-5)(-2+2)f(-2) + (-2)a+b
==> 2a - b = 5 ⋯⋯ (iv)
solving (iii), (iv), we get,
a = 2, b = -1
therefore, the remainder is (2x - 1).

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