MathDB
Nonzero Coefficient of Polynomials

Source: 2002 AMC12A Problem 25

August 2, 2011
algebrapolynomialAMC 8AMC 10AMC

Problem Statement

The nonzero coefficients of a polynomial PP with real coefficients are all replaced by their mean to form a polynomial QQ. Which of the following could be a graph of y=P(x)y = P(x) and y=Q(x)y = Q(x) over the interval −4≤x≤4-4\leq x \leq 4?
[asy]//Choice A size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125; } real g(real x) { return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(A)</span><span class='latex-bold'>(A)</span>", (-5,4.5)); [/asy] [asy]//Choice B size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2; } real g(real x) { return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(B)</span><span class='latex-bold'>(B)</span>", (-5,4.5)); [/asy] [asy]//Choice C size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.21875 x^2+0.28125 x+0.5; } real g(real x) { return -0.375 x^2-0.75 x+0.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(C)</span><span class='latex-bold'>(C)</span>", (-5,4.5)); [/asy] [asy]//Choice D size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875; } real g(real x) { return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625; } real z=3.14; draw(graph(f,-z, z), heavygray); draw(graph(g,-z, z), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(D)</span><span class='latex-bold'>(D)</span>", (-5,4.5)); [/asy] [asy]//Choice E size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598; } real g(real x) { return -0.166667 x^3+0.125 x^2+0.479167 x-0.375; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(E)</span>", (-5,4.5)); [/asy]
Back to ProblemsView on AoPS