f.4 Maths

When the polynomial 5x^3-4x^2+7 is divided by x^2-3, the remainder is?
5x^3-4x^2+7
=5x(x^2-3)+15x-4x^2+7
=5x(x^2-3)-4(x^2-3)-12+15x+7
=5x(x^2-3)-4(x^2-3)+15x-5
if the polynomial 5x^3-4x^2+7 is divided by x^2-3, the remainder
=15x-5


If the polynomial 9x^3+3x is divided by 3x+k, the remainder is k. Find the value of k.

let f(x)=9x^3+3x
f(-k/3)=k
-9k^3/27 -k=k
-k^3/3=2k
-k^3=6k
k^3+6k=0
k(k^2+6)=0
k=0 (k^2+6>0 for any value of k)


If x-2 is a factor of x^3-3x^2+ax+b, fidn the value of 2a+b-5.

let f(x)=x^3-3x^2+ax+b
f(2)=0
8-12+2a+b=0
2a+b-4=0
2a+b-5=-1


Let f(x)=x^3+ax^2+bx+5. If x+1 is a factor of f(x)-10 and f(2)=13, find the value of a+b.

f(x)-10=x^3+ax^2+bx-5
f(-1)-10=0
-1+a-b-5=0
a-b=6---------(1)
f(2)=13
8+4a+2b+5=13
4a+2b=0
2a+b=0-------(2)
(1)+(2)
3a=6
a=2
b=-2a=-4
a+b=2-4=6


When a polynomial P(x) is divided by 2x-5, the remainder is R. Find the remainder when P(x) is divided by 5-2x.

5-2x= -(2x-5)
the remainder is the same when P(x) is divided by 2x-5 or 5-2x
when P(x) is divided by 5-2x,the remainder= R

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Let f(x)= 9x3+3x.
Remainder=f(-k/3)
k=9(-k/3)3+3(-k/3)
k=k3/3 -k

3k=k3-3k
k3-6k=0
k(k2-6)=0
k=0 or k=√6


Let f(x)= x3-3x2+ax+b
∵x-2 is a factor of f(x)
∴f(2)=0
23-3(2)2+2a+b=0
8-12+2a+b=0
-4+2a+b=0
2a+b-5=-1