Please help with this polynomial long division problem.

[BlueStreak]

What is the value of k such that (x-k) is a factor of x³-x²+kx-30?

 

[tkhunny]

Why not actually perform the "long division"?

 

[pka]

What is the value of k such that (x-k) is a factor of x³-x²+kx-30?

If \((x-k)\) is a factor of the polynomial \(p(x)\) then it must be true that \(p(k)=0\). The Factor Theorem.

 

[BlueStreak]

@pka Does that mean I just replace all the x's with a zero?

 

[pka]

@pka Does that mean I just replace all the x's with a zero?

Frankly I did not read the title.
Replace \(x\) with \(k\), \(k^3-k^2+k(k)-30=0\) then solve for \(k\).

 

[BlueStreak]

@tkhunny I have tried long division, but I'm not entirely sure how to do that if I don't know one of the coefficients.

 

[BlueStreak]

@pka That would get me a decimal of about 3.107.

 

[pka]

@pka That would get me a decimal of about 3.107.

Sorry but I don't know how to use decimals. I know that \(k=\sqrt[3]{30}\) works.

 

[BlueStreak]

@pka Yeah, that's what I got. Thank you so much!

 

[Steven G]

If x-k is a factor then x=k is a root. So as pka suggested, replace x with k, k³−k²+k(k)−30=0 then solve for k.

 

[BlueStreak]

@Jomo Thank you so much! Yeah, I did that and got k = ³√30.

 

[Cubist]

I have tried long division, but I'm not entirely sure how to do that if I don't know one of the coefficients.

FYI: This is how the long division can be done...

                       x^2  +   (k-1)*x  +   k^2
       -----------------------------------------
  x-k  )    x^3     -  x^2  +       k*x      -30
            x^3     -k*x^2           .         .
          ----------------           .         .
                 (k-1)*x^2  +       k*x        .
                 (k-1)*x^2  - k*(k-1)*x        .
                 -----------------------       .
                                  k^2*x       -30
                                  k^2*x  -    k^3
                                  ---------------
                                           k^3-30   Remainder

If (x-k) is a factor then the remainder needs to be zero, therefore k^3-30=0

Much slower than pka's method!

 

[pka]

Much slower than pka's method!

Yes it is! It is also a perfect example of why there has been a radical drop in interest among the 'screen generation' is studying mathematics. I have put out a challenge to traditionalist to show a mathematics textbook teaching how one finds \(\sqrt{13}\) by hand.