Question

Which is the polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3 and roots \sqrt{5}and 2?
f(x)=3x3-6x2-15x+30
f(x)=x3-2x2-5x+10
f(x)=3x2-21x+30
f(x)=x2-7x+10

Answer 1

f(x)=3x3 - 6x2 - 15x+30

Answer 2

Leading coefficient means the coefficient (mulitplicative factor) of the highest power.
Only two of the above have a leading coefficient of 3, these being
3x3-6x2-15x+30 or 3x2-21x+30, with 3 being the leading coeff. of the third power and second power respectively.

Now we need our roots to fit. Two roots have been provided: sqrt(5) and 2.
We can use our knowledge of relationships between roots to do this.
Let's start with the easiest case, the quadratic.

3x2-21x+30 in the form ax2 + bx + c
The sum of roots is given by -b/a = -(-21)/3 = 7
Product of roots is given by c/a = 30/3 = 10

If the roots of this quadratic were sqrt(5) and 2 as given, they must fulfil the sum of roots and product of roots criteria. However they do not, since sqrt(5)+2 is not 7 and sqrt(5) x 2 is not 10.
Thus we are left with the cubic as our answer 3x3-6x2-15x+30

We can verify our roots by substituting x = sqrt(5) and x = 2 in and see they make the cubic equal to 0.

Alternatively you can use relationship of roots for cubics (sum and product of roots) to identify a third possible root being - sqrt(5), which together with sqrt(5) and 2 would satisfy the conditions required.

Answer 3

you can consider hit and trial as i did in my answer . when you see the option there is no option with any sqrt() term in this . so best guess is that third root is -sqrt(5) which will cancel out all the sqrt() terms