## sabika13 3 years ago The graph of f(x) = ax^4 + bx^2 + cx-24 crosses the x-axis at 1, -2, and 3. determine the equation of f(x).

1. satellite73

2. sabika13

do i expand and then solve?

3. satellite73

no you don't really have to expand yet, you can visualize what the constant should be the constant will be \(-6r\) so you know \(-6r=-24\) and if you are lucky \(r=4\) works

4. satellite73

oh damn it doesn't because you get a cube term and your polynomial doesn't seem to have one

5. satellite73

on the other hand something is wrong, because you have a 4th degree polynomial. it has 4 zeros total, although either two or four could be complex. however, if it crosses the \(x\) axis 3 times it must cross it a fourth. if one root is repeated, then it does not cross the axis but only touches it. so i am stumped

6. sabika13

maybe one passes through origin?

7. satellite73

then it still has 4 zeros one would be zero, and also it could not end in \(-24\)

8. satellite73

it really starts with \(ax^4\) right?

9. sabika13

yeah

10. sabika13

i have the answer if that helps?

11. satellite73

ok i am very certain that a fourth degree polynomial cannot cross the \(x\) axis only 3 times

12. satellite73

sure go ahead and let me see why i am wrong

13. sabika13

its f(x)= -4x^3 + 8x^2 + 20 x -20

14. satellite73

we could perhaps start with \[a(x-1)(x+2)(x-3)(x-r)\] and see if we can solve for the coefficients, but it must cross 4 times

15. sabika13

f(x)= -4x^3 + 8x^2 + 20 x -24 *** sorrry

16. satellite73

hold the phone, it is cubic?

17. sabika13

18. satellite73

i thought it was a 4th degree polynomial, since you wrote \[f(x) = ax^4 + bx^2 + cx-24\]

19. satellite73

if it is \[f(x) = ax^3 + bx^2 + cx-24\] then that is a whole different story

20. sabika13

lol thats what the question says, its ax^4 + bx^2 ..

21. satellite73

and much simpler write \[a(x-1)(x-3)(x+2)= ax^3 + bx^2 + cx-24\] and solve

22. satellite73

then there is something wrong with the question, it is a typo

23. sabika13

so if its ax^3 will the answer be right?

24. satellite73

your question asks for a 4th degree polynomial, which must have 4 zeros. it also has no cube term your answer has a third degree polynomial, so there is a mistake in the question

25. sabika13

yeah it prob is a typo.. ah been trying to question for so long.. thanks tho!