Cubic Functions Question

• May 26th 2012, 05:52 PM
Jacktenstar
Cubic Functions Question
Find the value for k, if the remainder is 12, when 2x^3+kx^2-18x+20 is divided by x-2.

I solved this question using long division and I got the correct answer of k=3.

My teacher says though I should do it this way:

P(2)=12

Therefore: 2(2)^3+k(2)^2-18(2)+20=12

16 +4k-36+20=12

4k=12

Therefore k=3

I understand that process of solving, but I don't understand how P(2) I get from x-2 =12(the remainder)

Anyone help explain??

Thanks

PS: Anyone know where I could find some worded problems for Cubic Functions, because we have not done any but there will be some in our Cubic Functions Test tomorrow?????
• May 26th 2012, 06:02 PM
emakarov
Re: Cubic Functions Question
Quote:

Originally Posted by Jacktenstar
I understand that process of solving, but I don't understand how P(2) I get from x-2 =12(the remainder)

Who said that x - 2 = 12? The teacher suggested using the polynomial remainder theorem.
• May 26th 2012, 06:14 PM
Jacktenstar
Re: Cubic Functions Question
I never said that x-2 =12, I said that I get P(2) from the x-2.

So does that mean P(x) which is in this case P(2)=remainder????
• May 26th 2012, 06:20 PM
emakarov
Re: Cubic Functions Question
Quote:

Originally Posted by Jacktenstar
So does that mean P(x) which is in this case P(2)=remainder????

Yes, P(x) is the remainder when P(y) is divided by (y - x).
• May 26th 2012, 06:24 PM
Jacktenstar
Re: Cubic Functions Question
Excuse me for my ignorance. What is P(y)? Does it mean the value of the second variable, in this case K???
• May 26th 2012, 06:37 PM
emakarov
Re: Cubic Functions Question
In mathematics, as well as in physics and programming, it is customary to distinguish lower-case and upper-case variable names, so k and K are generally not the same variable.

I tried to correctly formulate your statement "P(x) which is in this case P(2)=remainder." What is x here? The original problem formulation does not mention any specific x. Rather, x is a variable that ranges over the domain of the polynomial, i.e., all real numbers. Nowhere does the problem says that x = 2.

In the problem, P is a polynomial given by a formula. The formula may use any variable to range over the domain. This variable is specified in parentheses after the function name. Thus, P(x) = 2x^3 + kx^2 - 18x + 20 and P(y) = 2y^3 + ky^2 - 18y + 20 is the same polynomial. So, instead of the phrase "P(x) which is in this case P(2)=remainder," whose meaning is not clear because x is undefined, I said "(For all real numbers x,) P(x) is the remainder when P(y) is divided by (y - x)." This is the same as "For all real numbers x0, P(x0) is the remainder when P(x) is divided by (x - x0)" or "For all real numbers a, P(a) is the remainder when P(x) is divided by (x - a)."
• May 26th 2012, 06:44 PM
Jacktenstar
Re: Cubic Functions Question
Thanks. Would you know where I can find some worded questions for Cubic Functions online??

Thanks