? 發問於 科學及數學數學 · 1 十年前

# Two questions on polynomial (F.4)

1.When a polynomial f(x) is divided by x-5,the remainder is 9.When it is divided by x+2,the remainder is -5.Find the remainder when f(x) is divided by (X-5)(x+2).

2.Let f(x) =3x^3 +mx^2 -nx -7 .When f(x) is divided by (x+1)(x-3),the remainder is 2x-4.

(a)Find the values of f(-1) and f(3).

(b)Set up two equations connecting m and n.

(c)Find the values of m and n.

### 2 個解答

• 1 十年前
最愛解答

1.When a polynomial f(x) is divided by x-5,the remainder is 9.When it is divided by x+2,the remainder is -5.Find the remainder when f(x) is divided by (x-5)(x+2).

SOLUTION

let the remainder is ax+b

then

f(x)=g(x)(x-5)(x+2)+(ax+b)

Then by remainder theorem

f(5)=5a+b=9...(1)

f(-2)=-2a+b=-5...(2)

(1)-(2):

7a=14

a=2

sub into (2) b=2a-5=-1

So the remainder when f(x) is divided by (x-5)(x+2) is

2x-1

2.Let f(x) =3x^3 +mx^2 -nx -7 .When f(x) is divided by (x+1)(x-3),the remainder is 2x-4.

(a)Find the values of f(-1) and f(3).

(b)Set up two equations connecting m and n.

(c)Find the values of m and n.

SOLUTION

(a)

We have

f(x)=g(x)(x+1)(x-3)+2x-4

f(-1)=2(-1)-4=-6

f(3)=2(3)-4=2

(b)

Since f(-1)=-6

f(-1)=3(-1)^3 +m(-1)^2 -n(-1) -7=-6

-3+m+n-7=-6

m+n=4...(1)

This is the first equation

Since f(3)=2

f(3)=3(3)^3 +m(3)^2 -n(3) -7=2

81+9m-3n-7=2

9m-3n=-72

3m-n=-24

n-3m=24...(2)

This is the second equation

(c)

from (1)

m=4-n

sub into (2)

n-3(4-n)=24

-12+4n=24

n=9

m=4-9=-5

2007-04-18 00:33:48 補充：

答題中的 g(x) 就是 quotinet.應該用Q(x) 會好看些﹐不過其實都是同一樣東西。

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• 1 十年前

1)

Assume the remainder be Ax + B where A and B are constants.

i.e. f(x) = Q (x-5)(x+2) + (Ax + B)

As given, f(5) = 9 and f(-2) = -5

f(5) = 5A + B = 9 --- (1)

f(-2) = -2A + B = -5 --- (2)

(1) - (2)

7A = 14

A = 2

Put it back into (1)

5(2) + B = 9

B = -1

So, remainder is 2x - 1

**********************

2)

a) Assume f(x) = Q(x+1)(x-3) + (2x-4) where Q is the quotient.

f(-1) = 2(-1) - 4 = -6

f(3) = 2(3) - 4 = 2

b) f(x) = 3x^3 + mx^2 - nx - 7

f(-1) = 3(-1)^3 + m(-1)^2 - n(-1) - 7 = -6

-3 + m + n - 7 = -6

m + n = 4 --- (1)

f(3) = 3(3)^3 + m(3)^2 - n(3) - 7 = 2

81 + 9m - 3n - 7 = 2

3m - n = -24 --- (2)

c)

(1) + (2)

4m = -20

m = -5

n = 9

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