Factor theorem qn...?

Question

Given that kx^3+2x^2+2x+3 and kx^3-2x+9 have a common factor, what are the possible values of k?

This is one of the question from my assessment book, please help step by step if you know how to get to the answer..10PTS!!

Okay, thanks for replying so fast.
I just want to clarify 1 thing;
When a question say eg(x-a) is a factor of something, does it neccessarily mean that f(x)=0 everytime? or it may have a remainder?

Answer 1

Let f(x) = kx^3 + 2x^2 + 2x + 3 and g(x) = kx^3 - 2x + 9.
Let the common factor of f(x) and g(x) be (x - a).

By Factor Theorem,
f(a) = g(a) = 0
ka^3 + 2a^2 + 2a + 3 = ka^3 - 2a + 9 = 0
2a^2 + 4a - 6 = 0
a^2 + 2a - 3 = 0
(a + 3)(a - 1) = 0
a = -3 or a = 1

Substituting values of a into f(x)......
when a = -3,
f(-3) = 0
k(-3)^3 + 2(-3)^2 + 2(-3) + 3 = 0
-27k + 18 - 6 + 3 = 0
k = 5/9

when a = 1,
f(1) = 0
k(1)^3 + 2(1)^2 + 2(1) + 3 = 0
k + 2 + 2 + 3 = 0
k = -7

Alternatively, substituting values of a into g(x)......
when a = -3,
g(-3) = 0
k(-3)^3 -2(-3) + 9 = 0
-27k + 6 + 9 = 0
k = 5/9

when a = 1,
g(1) = 0
k(1)^2 -2(1) + 9 = 0
k - 2 + 9 = 0
k = -7

Therefore, the possible values of k are 5/9 and -7.

Edit: This is in reply to the question you wanted to clarify.
When a question states that (x - a) is a factor of f(x), the remainder of f(x) when divided by (x - a) must certainly be equals to 0, which means that f(a) = 0. This is defined as Factor Theorem.

Answer 2

Any common factor of these 2 polynomials
must divide their difference.
So it divides
2x² + 4x -6
or 2(x² + 2x-3) = 2(x+3)(x-1).
Suppose that x-1 is a factor of kx³ + 2x² + 2x +3.
By long division or synthetic division we find that
the remainder is k+7, so k = -7 is a possible answer.
Also we find that -7x³ -2x +9 has x-1 as a factor.
Similarly if x+3 is a factor of kx³ + 2x² + 2x+3
we find, again by division, that the remainder is
-27k + 15 = 0
So k = 5/9 and k = 5/9 makes kx³ -2x +9
have a factor of x+3 also.
So the 2 possible values of k are k = -7 and k = 5/9.