Question

If α and β are the zeros of the quadratic polynomial f(x) = 3x2–4x + 5, find a polynomialwhose zeros are 2α + 3β and 3α + 2β.

Answer 1

`\boxed{\sf \ \ \ 3x^2-20x+37\ \ \ }`

Explanation:

Hello,

a and b are the zeros, we can say that

`f(x)=3(x^2-(4)/(3)x+(5)/(3)) = 3(x-a)(x-b)=3(x-(a+b)x+ab)`

So we can say that

`a+b=(4)/(3)\\ab=(5)/(3)`

Now, we are looking for a polynomial where zeros are 2a+3b and 3a+2b

for instance we can write

`(x-2a-3b)(x-3a-2b)=x^2-(2a+3b+3a+2b)x+(2a+3b)(3a+2b)\\= x^2-5(a+b)x+6a^2+6b^2+9ab+4ab`

and we can notice that

`a^2+b^2=(a+b)^2-2ab` so

`(x-2a-3b)(x-3a-2b)=x^2-5(a+b)x+6[(a+b)2-2ab]+13ab\\= x^2-5(a+b)x+6(a+b)^2+ab`

it comes

`x^2-5*(4)/(3)x+6((4)/(3))^2+(5)/(3)`

multiply by 3

`3x^2-20x+2*16+5=3x^2-20x+37`