Try this beautiful Problem based on Vieta's Formula from AMC 10A, 2021 Problem 14.
All the roots of the polynomial $z^{6}$-$10 z^{5}$+$A z^{4}$+$B z^{3}$+$C z^{2}$+$D z+16$ are positive integers, possibly repeated. What is the value of $B$ ?
Vieta's Formula
Polynomial
Roots of the polynomial
Problem-Solving Strategies by Arthur Engel
AMC 10A 2021 Problem 14
-88
Find out the degree of the given polynomial.
We know, Degree of polynomial= Number of roots of that polynomial.
Apply Vieta's Formula on the given polynomial.
By Vieta's Formula, the sum of the roots is 10 and product of the roots is 16.
Since there are 6 roots for this polynomial. By trial and check method find the roots.
The roots should be $2, 2, 2, 2, 1, 1$.
Now using the roots reconstruct the polynomial.
So the polynomial should be -
$(z-1)^{2}(z-2)^{4}$
$=(z^{2}-2 z+1)\\(z^{4}-8 z^{3}+24 z^{2}-32 z+16)$
Now equate it with the given polynomial to find the value of $B.$
Try this beautiful Problem based on Vieta's Formula from AMC 10A, 2021 Problem 14.
All the roots of the polynomial $z^{6}$-$10 z^{5}$+$A z^{4}$+$B z^{3}$+$C z^{2}$+$D z+16$ are positive integers, possibly repeated. What is the value of $B$ ?
Vieta's Formula
Polynomial
Roots of the polynomial
Problem-Solving Strategies by Arthur Engel
AMC 10A 2021 Problem 14
-88
Find out the degree of the given polynomial.
We know, Degree of polynomial= Number of roots of that polynomial.
Apply Vieta's Formula on the given polynomial.
By Vieta's Formula, the sum of the roots is 10 and product of the roots is 16.
Since there are 6 roots for this polynomial. By trial and check method find the roots.
The roots should be $2, 2, 2, 2, 1, 1$.
Now using the roots reconstruct the polynomial.
So the polynomial should be -
$(z-1)^{2}(z-2)^{4}$
$=(z^{2}-2 z+1)\\(z^{4}-8 z^{3}+24 z^{2}-32 z+16)$
Now equate it with the given polynomial to find the value of $B.$
