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Jamie and Billy each think of a polynomial. Each of their polynomials is monic, has degree 3, and has the same positive constant term and the same coefficient of x. The product of their polynomials is  x^6 +2x^5 +2x^4 +18x^3 + 33x^2+40x+81.  What is the constant term of Jamie's polynomial?

 Apr 10, 2021
 #1
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Let the constant term of the 3rd degree polynomial equal a. Since the constants in the 3rd degree polynomial multiply to equal the constant in the 6th-degree polynomial, \(a^2=81\). Since none of the signs in the 6th-degree polynomial are negative, the polynomials of Jamie and Billy must both have all negative signs or all positive signs, but since it specified monic polynomials, then it must be that all of them are positive, so the answer is just \(\boxed{9}\)

 Apr 10, 2021
 #2
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y=ax³+bx²+cx+d
the constant term is d


if their polynomials have the same constant term  then the product is
(ax³+bx²+cx+d)(rx³+ex²+vx+d)
the only thing we know is the constant terms are the same
we notice that the last term of the product will be the product of the constant terms

so the last term will be d²

the last term is 16
d²=16
sqrt both sides
d=4

the constant term of Jamie's polynomial is 4

 Apr 10, 2021

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