1. ## URGENT - algebra

f(x) = xcubed - 2xsquared + ax + b, where a and b are constants.
when f(x) is divided by (x-2), the remainder is 1.
when f(x) is divided by (x+1), the remainder is 28.

a) Find the value of a and the value of b.
b) Show that (x-3) is a factor of f(x)

Thanks a lot!

2. This is about the factor and remainder theorem: the remainder when f(x) is divided by (x-a) is just f(a). In this case, you're being told f(2)=1, f(-1)=28. That means that 2^3 - 2.2^2 + 2a + b = 1 and (-1)^3 - 2.(-1)^2 - a + b = 28. This is a pair of simultaneous linear equations in a and b, namely 2a+b=1, b-a=31, with solution a=-10, b=21. Now you're asked about f divided by (x-3), so look at f(3) = 3^3 -2.(3^2) -10.3 + 21 = 0. So the remainder on division by (x-3) is zero, and f is divisible by x-3.

3. thanks a lot!!

4. Originally Posted by devilicious

f(x) = xcubed - 2xsquared + ax + b, where a and b are constants.
when f(x) is divided by (x-2), the remainder is 1.
when f(x) is divided by (x+1), the remainder is 28.

a) Find the value of a and the value of b.
b) Show that (x-3) is a factor of f(x)

Thanks a lot!
Hello,

I presume that you know how to do long division:

$\displaystyle \left( x^3-2x^2+ax+b \right)/(x-2)=x^2+a \ remainder\ b+2a$

$\displaystyle \left( x^3-2x^2+ax+b \right)/(x+1)=x^2-3x+(a+3)$. The remainder is here: b-a-3

According to the text of your problem you'll get a system of two linear equations:
$\displaystyle \left\{\begin{array}{cc}2a+b=1\\-a+b-3=28\end{array}\right.$

You get a = -10 and b = 21. So your equation now reads:
$\displaystyle f(x)=x^2-2x^2-10x+21=(x-3) \cdot (x^2+x-7)$, which shows that f(x) is divisible by (x-3).

Greetings

EB