Question

Use the Rational Zeros Theorem to write a list of all possible rational zeros of the function.

f(x) = -2x4 + 4x3 + 3x2 + 18

Answer 1

two key numbers here: the leading coefficient, and the constant.

What are they?

Answer 2

i have no clue.

Answer 3

Leading coefficient is the number multiplied to the x with the *highest exponent* while the constant, is, well, the constant (the number with no x attached to it)

So... what are the leading coefficient and the constant (respectively)?

Answer 4

2x^4 and 18?

Answer 5

drop the x.
And it's actually -2.

Answer 6

Now, here's what needs to be done...
I want you to list down all the factors of 2 and also all the factors of 18.

Answer 7

2
2,1

18
1,18
2,9
3,6

Answer 8

There really is no need to pair them up :)
But anyway, well done :P

So, to get the *possible* rational root, just pair them up as fractions, with the factors of 18 in the numerator and factors of 2 in the denominator.

In other words, the possible rational roots take the form
\[\Large \pm\frac{q}p\]
Where q is a factor of 18 and p is a factor of 2.

Don't be redundant, though... there is no need to include, say, both
\[\Large \pm\frac{6}2\color{orange}{=3} \text{ and} \ \pm \frac{3}{1} \color{orange}{=3}\]
Since they are obviously the same

Answer 9

im confused..

Answer 10

so like 9/2 2/1 ?

Answer 11

Just list down all possible pairings of a factor of 18 and a factor of 2.
For example, 6 (factor of 18) and 2 (factor of 2)

And just divide them like so: (don't forget the plus-minus)
\[\Large \pm\frac63= \pm 2\]

For example, say the leading coefficient was 2 and the constant was 6.
factors of 2 = {1,2}
factors of 6 = {1,2,3,6}

then, the possible rational roots would be all possible divisions between factors of 6 and 2.
\[\Large PRR = \left\{\pm\frac61 , \pm\frac62,\pm\frac21, \pm\frac32,\pm\frac22, \pm\frac12\right\}\]\[\Large PRR = \left\{\pm 6,\pm 3, \pm 2\pm \frac32,\pm1,\pm\frac12\right\}\]

I didn't include 3/1 because it's the same as 6/2
I didn't include 1/1 because it's the same as 2/2

But that's all of it :)

Answer 12

Now, your constant is 18 (not 6) so expect more possible rational roots... but the essence remains the same :D

Good luck ^_^

Answer 13

ohhh thank you so much!

Answer 14

No problem ^_^

Answer 15

Post your answer if you can.