[MOVED] Polynomial degrees and zeros (chose answer)

[bleach_bankai]

Find a polynomial of degree 3 whose zeros are -3, 3/2 and 2

a.2x^3-15x-18
b.2x^2+3x-9
c.2x^2-7x+6
d.2x^3-x^2-15x+18
e.2x^3-7x^2-15x+18

I know C and B are not options because the degree is not three, but how do I figure out the rest?

 

[wjm11]

Re: Polynomial degrees and zeros

Find a polynomial of degree 3 whose zeros are -3, 3/2 and 2

a.2x^3-15x-18
b.2x^2+3x-9
c.2x^2-7x+6
d.2x^3-x^2-15x+18
e.2x^3-7x^2-15x+18

One way is to plug in the three given values and see if the function equals zero.

Another way is to say the function must be

y = k(x-(-3))(x-(3/2))(x-2) = (x+3)(2x-3)(x-2)

You can multiply this out and see of it matches any of the offered solutions.

 

[soroban]

Hello, bleach_bankai!

Another approach . . .

\(\displaystyle \text{Find a polynomial of degree 3 whose zeros are: }\, \text{-}3,\:\tfrac{3}{2},\text{ and }2\)

. . \(\displaystyle \begin{array}{cc}(a)& 2x^3-15x-18 \\ (b) & 2x^2+3x-9 \\ (c) & 2x^2-7x+6 \\ (d) & 2x^3-x^2-15x+18 \\ (e) & 2x^3-7x^2-15x+18 \end{array}\)

There is a theorem available for this situtation.
. . I'll modify it specifically for this problem.


\(\displaystyle \text{Given a cubic equation: }\:ax^3 + bx^2 + cx + d \:=\:0\)

. . \(\displaystyle \text{divide by the leading coefficient: }\:x^3 + \tfrac{b}{a}x^2 + \tfrac{c}{a}x + \tfrac{d}{a} \:=\:0\)

\(\displaystyle \text{Then: }\:\begin{Bmatrix}\text{sum of the roots} &=& -\frac{b}{a} \\ \\[-3mm] \text{product of the roots} &=& -\frac{d}{a} \end{Bmatrix}\)



\(\displaystyle \text{In our problem: }\:\begin{Bmatrix}\text{sum of roots} &=&\text{-}3+\frac{3}{2}+2 &=& \frac{1}{2} \\ \\[-3mm] \text{product of roots} &=& (\text{-}3)(\frac{3}{2})(2) &=& \text{-}9 \end{Bmatrix}\)


\(\displaystyle \text{The only equation with }\:\tfrac{b}{a} = \text{-}\tfrac{1}{2}\:\text{ and }\:\tfrac{d}{a} = 9\:\text{ is: }\:2x^3 - x^2 - 15x + 18\) .(d)

 

[Deleted member 4993]

There is a brute-force method:

Evaluate all those functions at the given roots (-3, 3/2 and 2) and find out which one of those functions satisfy the condition.

If you use a "spreadsheet" this method would be fastest - however during exam you may not have access to a spreadsheet program.