Question

Please Help...
Find the expression for a cubic function (f) if f(1)=6 and f(-1)= f(0)= f(2)= 0

Answer 1

\[\large f(-1)=f(0)=f(2)=\huge \color{goldenrod}{0}\]

See how they all equal 0? That's very important.
It tells us that each of those x values are ROOTS of our polynomial.

What can we do with a root? Welllll, we can rewrite it as a FACTOR of our polynomial.

\(\large f(-1)=0\) from this we can say that \(\large x=-1\) is a root of our polynomial.
Adding 1 to both sides gives us, \(\large x+1=0\).

This is one of our FACTORS of our polynomial that we're trying to find.
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Let's do the same with the other roots.
\(\large f(0)=0\) this one tells us that \(\large x=0\) is a root.
In other words, \(\large x\) is a factor of our polynomial.
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\(\large f(2)=0\) tells us that \(\large x=2\) is a root.
Subtracting 2 from each sides gives us another factor, \(\large x-2\).

Answer 2

If we put these factors togetherrrrrrrr, we get,\[\large f(x)=(x)(x-2)(x+1)+C\]We'll have to use the extra information \(\large f(1)=6\) to solve for C.

Answer 3

\[\large f(1)=6 \qquad \rightarrow \qquad 6=(1)(1-2)(1+1)+C\]Solving for C gives us,\[\large C=8\]

Answer 4

From there, it's probably a good idea to put the cubic in standard form.
Expand out those brackets.

Answer 5

thank you. im still unsure of exactly what I'm doing and the answer in the book is different than what you have and i dont know how to arrive at the answer in my book. the answer that i have here is \[f(x)= -3x(x+1)(x-2)\]

Answer 6

That's the answer? ok ok lemme see what I did wrong c: one sec

Answer 7

Ok I guess I was mistaken. You don't want to throw a +C on the end.
Your polynomial will actually be MULTIPLIED by an unknown constant.\[\large \cancel{f(x)=x(x-2)(x+1)+C} \qquad f(x)=Cx(x-2)(x+1)\]Plug in our information to solve for C,\[\large f(1)=6 \qquad \rightarrow \qquad 6=C(1)(1-2)(1+1)\]\[\large 6=C(-2) \qquad \rightarrow \qquad C=-3\]

Answer 8

Which gives us,\[\large f(x)=-3x(x-2)(x+1)\]

Answer 9

Hmmm this is a bit of a tricky problem :O

Answer 10

thank you so much for your help!