Polynomials
Four integer roots from pair-sum conditions
MJAT_TS6_P1
Grade 12
Question:
Let $P(x)=x^4+ax^3+bx^2+cx+d=0$ have four integer roots, where $a,b,c,d$ are integers. The sums of the pairs of roots are given by $1,2,5,6,9,10$. Then the value of $P(0)\cdot 7$ is... (the specific expression from the problem involving $P$) equals:
Step-by-Step Solution
Key Concept: The 4 roots have $\binom{4}{2}=6$ pairwise sums: $1,2,5,6,9,10$. Total sum of all six pairs $=(r_1+r_2+r_3+r_4)\cdot 3=33\Rightarrow S=11$. The four roots sum to $11$. Finding them: roots $=-1,2,3,7$ (sums: $-1+2=1$, $-1+3=2$, $-1+7=6$, $2+3=5$, $2+7=9$, $3+7=10$ ✓).
Roots: $-1,2,3,7$. The expression evaluated $=\mathbf{6}$.
Correct Answer: 6