MathDB
Two Polynomials

Source: 2017 AMC10A #24, 2017 AMC12A #23

February 8, 2017
AMCAMC 10AMC 12algebrapolynomial2017 AMC 10A2017 AMC 12A

Problem Statement

For certain real numbers aa, bb, and cc, the polynomial g(x)=x3+ax2+x+10g(x) = x^3 + ax^2 + x + 10 has three distinct roots, and each root of g(x)g(x) is also a root of the polynomial f(x)=x4+x3+bx2+100x+c.f(x) = x^4 + x^3 + bx^2 + 100x + c. What is f(1)f(1)?
<spanclass=latexbold>(A)</span> 9009<spanclass=latexbold>(B)</span> 8008<spanclass=latexbold>(C)</span> 7007<spanclass=latexbold>(D)</span> 6006<spanclass=latexbold>(E)</span> 5005<span class='latex-bold'>(A)</span>\ -9009 \qquad<span class='latex-bold'>(B)</span>\ -8008 \qquad<span class='latex-bold'>(C)</span>\ -7007 \qquad<span class='latex-bold'>(D)</span>\ -6006 \qquad<span class='latex-bold'>(E)</span>\ -5005