Can some one help me understand this. I need to help my little brother
2 and -1/3 are zeros of the polynomial \(\sf 3x^3-2x^2-7x-2\). Find the third zero.

- Abhisar

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- ParthKohli

Ninth mein hai ya tenth mein?

- Abhisar

9th me

- ParthKohli

Plugging @Jhannybean for "teh lulz".
The sum of roots is \(\frac{2}{3}\) so the third root is \(\frac{2}{3}-\left(2 -\frac{1}{3}\right)\)

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## More answers

- ParthKohli

Thus the answer is -1.

- ParthKohli

We could also have done it using the product of roots. The product of roots is \(2/3\) so the third root is\[\frac{2/3}{2\cdot (-1/3)} = \boxed{-1}\]

- ParthKohli

Oh, ninth mein... hmm. I don't think they teach that in ninth - it's taught only in tenth. Here's another method!

- Abhisar

Yes..

- ParthKohli

As I recall, they teach the so-called Factor Theorem in 9th grade. It says that if \(k\) is a root of a polynomial, then \(x-k\) must divide it. We know that \(2, -1/3\) are roots. Let the third root be \(k\). Then the polynomial can be written as\[3(x-2)(x+1/3)(x-k) = 3x^3 - 2x^2 - 7x - 2\]

- ParthKohli

Now comes the interesting part. Is he trying to understand the solution for himself or for his school-exams (as he has to write in his 9th exam-sheet)?

- Abhisar

Exams..

- Abhisar

Is that it? o_O

- ParthKohli

No. That's not it.\[3(x-2)(x+1/3)(x-k) = 3x^2 - 2x^2 - 7x - 2\]\[\Rightarrow x-k = \frac{3x^2 - 2x^2 - 7x - 2}{3(x-2)(x+1/3)}\]

- ParthKohli

\[x -k = \frac{3x^3 - 2x^2 - 7x - 2}{(x-2)(3x+1)} = \frac{3x^3 - 2x^2 - 7x - 2}{3x^2 - 5x - 2}\]

- ParthKohli

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- ParthKohli

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- ParthKohli

\[x-k = x+1\]\[k = -1\]

- Abhisar

Ok, I get it but why have you multiplied
\[(x-2)(x+\left(\begin{matrix}1 \\ 3\end{matrix}\right))(x-k)\] by 3

- ParthKohli

Good question - we have to do that to get a 3 on the \(x^3\) term. Otherwise the cubic expression still has the same roots, but will be \(x^3 - 2/3 x^2 - 7/3 x - 2/3\).

- ParthKohli

If you have known roots say \(a_1, a_2, a_3\) and you want to know the rest of the known roots of a given polynomial, it would make more sense to divide the polynomial by \((x-a_1)(x-a_2)(x-a_3)\). What this does is it reduces the degree of a polynomial and small-degree polynomials are always easier to solve.

- Abhisar

Okay....I get it...give me a min..

- Abhisar

All right I got it. Thanks a bunch. c:

- ParthKohli

Ussey samajh aaya?

- Abhisar

Ab samjhaunga na usko c:

- Abhisar

So 3 se multiply kiya taaki fraction me na rahe, right?

- ParthKohli

As a personal exercise, instead of long division, try expanding. \[3(x-2)(x+1/3)(x-k) = 3x^3 - 2x^2 - 7x - 2 \]\[=(x-2)(3x+1)(x-k) \]\[= 3x^3 + (-5 - 3k ) x^2 +(-k+6k-2)x +2k \]

- ParthKohli

Now compare coefficients:\[-5 - 3k = -2\]\[5k -2 = -7\]\[2k = -2\]All three equations tell you only one thing :)

- ParthKohli

Basically if you uniquely want to determine a polynomial function, you need to need the roots, but you ALSO need to know the leading coefficient.\[(x-1)(x-2)\]\[2(x-1)(x-2)\]\[3(x-1)(x-2)\]\[4(x-1)(x-2)\]\[\vdots \]all of these have roots \(1,2\). If you expand each, you'll see that the difference is the leading coefficient (actually all coefficients, but the importance of the leading coefficient is that it's the same as what has been multiplied outside)\[x^2 - 3x + 2\]\[2x^2 - 6x + 4\]\[3x^2 - 9x + 6\]\[4x^2 - 12x + 8\]

- ParthKohli

do you see how the leading coefficient (coefficient of highest power) is the same as the number that is multiplied outside now?

- ParthKohli

which is why\[(x-2)(x+1/3)(x-k) \]cannot equal \(3x^2 - 2x^2 +5x -2\) unless we multiply it by 3. it's because the leading coefficient and the number multiplied outside must be the same!

- ParthKohli

*Manmohan Singh shtyle*
Theek hai?

- Abhisar

Haha

- Abhisar

Ok, just to be sure.
Suppose 2x+1 is a factor of \(\sf 2x^2-x-1\) then other zero will be +1 ?

- ParthKohli

Yes.

- Abhisar

Ok, thanks again..

- ParthKohli

Badiya.

- Jhannybean

*

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