Thread: 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?????

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.

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????

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).

Excuse me for my ignorance. What is P(y)? Does it mean the value of the second variable, in this case K???

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)."

Thanks. Would you know where I can find some worded questions for Cubic Functions online??

Thanks