1. A quadratic polynomial $f(x)=x^{2}+p x+q$, with $p$ and $q$ real numbers, is said to be a double-up polynomial if it has two real roots, one of which is twice the other.
(a) If a double-up polynomial $f(x)$ has $p=-15$, determine the value of $q$.
(b) If $f(x)$ is a double-up polynomial with one of the roots equal to 4 , determine all possible values of $p+q$.
(c) Determine all double-up polynomials for which $p+q=9$.
Solution:
(a) Suppose a double-up polynomial has two roots which are $k$ and $2 k$. Then the polynomial is $(x-k)(x-2 k)=x^{2}-3 k+2 k^{2}$. When $p=-15$, we have $k=5$ so $q=2 \cdot(5)^{2}=50$.
(b) From the previous part, we know that $p+q=2 k^{2}-3 k$. If one root is equal to 4 then from above we either have $k=4$ or $k=2$. When $k=4$ we get $p+q=32-12=20$ and when $k=2$ we get $p+q=8-6=2$.
(c) When $p+q=9$ we have $2 k^{2}-3 k=9$ which has solutions $k=3, k=-\frac{3}{2}$. We get $x^{2}-9 x+18$ and $x^{2}+\frac{9}{2} x+\frac{9}{2}$ as the polynomials.