I am not sure how to go about the following problem:
Find the cubic polynomial function with two of its zeros: 2 and (-3+sqrt2) , and a y-intercept of 7.
This is what I did: y= a(x-2)(x+3-sqrt2) 7=a (0-2)(0+3-sqrt2) a= 7/(-2*(3-sqrt2)) a=-2.21 so my equation would be: y= -2.21x(x-2)(x+3-sqrt2)
However, the correct answer is y= -0.5(x-2)(x^2+6x+7). Where did I go wrong?
On
Your answer is not correct because the value of your polynomial at $x=0$ is $0$, not $7$. The question is not correct in asking for "the cubic polynomial" with these characteristics. Your start of $(x-2)(x+3-\sqrt 2)$ is correct. Now you need to multiply by one more linear term to get a cubic polynomial, but the one linear term you can't use is $x$ because that puts a root at $x=0$. With any other linear term you could then multiply by an overall constant to make $y(0)=7$. You needed to do the scaling after multiplying by the linear term. The author clearly wanted rational coefficients, so selected the conjugate linear term to $x+3-\sqrt 2$ to clear the square root. That was not required by the problem statement. One final point would be that your overall multiplicative factor should be exact, not something like $2.21$
So you can put $$y=a(x-b)(x-2)(x+3-\sqrt 2)$$ where $b$ is the third root and $a$ is constant. This has the two roots you are given and there is one further condition. This one condition is insufficient to determine the two remaining unknowns.
Given the text, there are plausible contexts in which it would be implied that either (1) $a=1$ or (2) $b=-3-\sqrt 2$
The "official answer" suggests that the context should imply (2).
Your own answer takes $b=0$, though your calculation seems to involve a quadratic which has the implication $b=-1$. This inconsistency invalidates your work. [This comment added after reading Ross Millikan's answer]
You also fail to give an exact expression for $a$, which I think would be obligatory given the form in which the roots are given (show you know how to bring the expression for $a$ into a rational expression involving $\sqrt 3$), rather than an approximation to two decimal places.
So even if you are right about the context, you could be marked wrong. You may be right in suggesting that the question is poorly posed. But if it is in a chapter or section on polynomials with rational coefficients you would be wrong.