Question

what are the rational zeros of \[6x ^{4}+32x ^{3}-70x ^{2}\]

Answer 1

\[f(x) = 6x^4 + 32x^3 - 70x^2\]
Let's start by factoring out the GCD, greatest common divisor
For example, everything up here is divisible by 2x^2, right?

Answer 2

yeah. that's what i was thinking of doing first, but now i don't know what to do.

Answer 3

so i would get an answer of 2x^2(3x^2+16x-35)?

Answer 4

That would be our first factorization, yes

Answer 5

Okay. So what would I do next?

Answer 6

\[2x^2(3x^2+16x-35)\]
So we at least know it has a zero at x = 0, but we might have rational zeros of that quadratic. Just to make sure: The question asked for all _actual_ rational zeros, right? Not just _possible_ rational zeros?

Answer 7

The exact question is: Find all the rational zeros of the function...

Answer 8

K gotcha! So now we know we have a rational zero of x = 0, but we might get more from that quadratic in parenthesis, right?

Answer 9

Yep.

Answer 10

\[2x^2 = 0\]
\[x = 0\]
\[3x^2 + 6x -35 = 0\]

Answer 11

So we can either (attempt to) factor, or use the quadratic formula

Answer 12

Do you think that doing the quadratic formula is easiest?

Answer 13

And I just went for computing the discriminant, to see if it could be factored: 6^2 - 4(3)(-35), which wound up being 676, which is indeed 26^2

Answer 14

16^2, not 6^2, I meant, sorry

Answer 15

I do, yeah. Otherwise we'd be looking for a pair of numbers that multiply to -105 that add up to 16, or "guess and check", and that all seems harder than just the quadratic formula

Answer 16

Okay.

Answer 17

\[b^2 - 4ac : 16^2 - 4(3) (-35) =676 = 26^2\]

Answer 18

So for my final answers I would get, x= 0, -1.66, and 7?

Answer 19

Nope. Rational zeros: 0, -8 and 5/3

Answer 20

How? I'm confused....

Answer 21

\[x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a }\]
\[= \frac{ -16\pm \sqrt{26^2} }{ 2(3) }\]
\[=\frac{ -16\pm 26 }{ 6 }\]
\[x = \frac{ -48 }{ 6 } = -8\]
\[x = \frac{ 10 }{ 6 } = \frac{ 5 }{ 3 }\]

\[Therefore \space \space rational \space \space zeros : 0, \space -8, \space \frac{ 5 }{3 }\]

Answer 22

Oh!!! I missed the negative on the 16. But I'm still confused about how you got -8. I keep getting -42 instead of -48.

Answer 23

\[\frac{ -48 }{ 6 } = -8\]

Answer 24

Isn't -16-26=-42 though? Or am I just confused.....