Question

$k(x) = x^4 - 14x^3 + 69x^2 - 140x + 100$
$g(x) = 2x^4 + 4x^3 + 4x^2 + 4x + 2$
$h(x) = x^4 + 11x^3 + 42x^2 + 64x + 32$
$f(x) = x^4 + 4x^2 + 3$

Explanation:

Step1: Analyze $f(x)=x^4+4x^2+3$

This is an even function ($f(-x)=f(x)$), so its graph is symmetric about the y-axis. It has no real roots, since $x^4+4x^2+3=(x^2+1)(x^2+3)$, and $x^2+1>0$, $x^2+3>0$ for all real $x$. The y-intercept is $f(0)=3$. This matches the bottom-right graph (symmetric over y-axis, no x-intercepts, y-intercept at 3).

Step2: Analyze $k(x)=x^4-14x^3+69x^2-140x+100$

Factor the polynomial: $k(x)=(x-2)^2(x-5)^2$. It has repeated real roots at $x=2$ and $x=5$, so the graph touches the x-axis at these points and opens upwards (leading coefficient positive). This matches the top-left graph (touches x-axis at 2 and 5, symmetric around $x=3.5$).

Step3: Analyze $h(x)=x^4+11x^3+42x^2+64x+32$

Factor the polynomial: $h(x)=(x+1)(x+2)^2(x+4)$. It has a real root at $x=-1$, a repeated root at $x=-2$, and a root at $x=-4$. The leading coefficient is positive, so as $x\to\pm\infty$, $h(x)\to+\infty$. The graph touches the x-axis at $x=-2$ and crosses at $x=-1, x=-4$. This matches the top-right graph (x-intercepts at -4, -2, -1, touches at -2).

Step4: Analyze $g(x)=2x^4+4x^3+4x^2+4x+2$

Factor the polynomial: $g(x)=2(x^2+1)^2$. It is an even function (symmetric about y-axis) with no real roots, since $x^2+1>0$ for all real $x$. The y-intercept is $g(0)=2$, and the leading coefficient is positive. This matches the bottom-left graph (symmetric over y-axis, no x-intercepts, y-intercept at 2).

Answer:

• $k(x)=x^4-14x^3+69x^2-140x+100$ → Top-left graph
• $h(x)=x^4+11x^3+42x^2+64x+32$ → Top-right graph
• $g(x)=2x^4+4x^3+4x^2+4x+2$ → Bottom-left graph
• $f(x)=x^4+4x^2+3$ → Bottom-right graph