The question is:
Find a cubic polynomial $p(x)$ whose zeroes are the same as those collectively of polynomials $g(x) = 2x^2 – 9x + 4$ and $f(x) = 2x^2 + 3x – 2$. Given that $p(0)$ = 8.
I tried solving the question but I got a little confused at the "collectively" part and also on how to use the value of $p(0)$.
Using the quadratic formula, I calculated the roots of $g(x)$ as 4 and 1/2.
Similarly, the roots for $f(x)$ came out to be 1/2 and -2.
And then I added them up to get two roots of the cubic polynomial $p(x)$ as 9/2 and -3/2.
( because they said it's roots are same as those collectively of the two quadratic polynomials. )
I don't exactly know how to proceed after this.
Please explain. Thanks.
Best Answer
Hint
We have
$$p(x)=\lambda(x-4)(x+2)\left(x-\frac12\right)$$ where $$p(0)=\lambda\times(-4)\times 2\times\left(-\frac12\right)$$