Question

question
use technology to find points and then graph the following polynomial, then write it in factor
$f(x) = x^4 + 4x^3 - 28x^2 - 64x + 192$
plot all of the points that fit on the axes.
you may click a plotted point to delete it.

Explanation:

Step1: Find roots of the polynomial

We can use a graphing calculator or software (like Desmos, TI - 84) to find the x - intercepts of the polynomial \(f(x)=x^{4}+4x^{3}-28x^{2}-64x + 192\). By analyzing the graph or using the rational root theorem (possible rational roots are factors of 192 divided by factors of 1, so \(\pm1,\pm2,\pm3,\pm4,\pm6,\pm8,\pm12,\pm16,\pm24,\pm32,\pm48,\pm64,\pm96,\pm192\)). We find that \(x = 2\), \(x=-6\), \(x = 4\), \(x=-4\) are roots of the polynomial.

Step2: Write in factored form

If \(r\) is a root of a polynomial \(f(x)\), then \((x - r)\) is a factor of \(f(x)\). So, since the roots are \(x = 2\), \(x=-6\), \(x = 4\), \(x=-4\), the factored form of the polynomial is \(f(x)=(x - 2)(x + 6)(x - 4)(x + 4)\). We can also expand the factored form to check:
\((x - 2)(x + 6)=x^{2}+4x-12\)
\((x - 4)(x + 4)=x^{2}-16\)
Then \((x^{2}+4x - 12)(x^{2}-16)=x^{4}-16x^{2}+4x^{3}-64x-12x^{2}+192=x^{4}+4x^{3}-28x^{2}-64x + 192\), which matches the original polynomial.

Step3: Graphing the polynomial

To graph the polynomial, we can use the roots to find key points. The x - intercepts are at \((2,0)\), \((-6,0)\), \((4,0)\), \((-4,0)\). We can also find the y - intercept by setting \(x = 0\), \(f(0)=0^{4}+4\times0^{3}-28\times0^{2}-64\times0 + 192=192\), so the y - intercept is \((0,192)\). We can also find some additional points by plugging in values of \(x\) (e.g., \(x = 1\), \(f(1)=1 + 4-28-64 + 192=105\), so \((1,105)\); \(x=-1\), \(f(-1)=1-4 - 28 + 64+192=225\), so \((-1,225)\)) and then plot these points and draw a smooth curve through them, noting the end - behavior (as \(x
ightarrow\pm\infty\), \(x^{4}\) dominates, so \(f(x)
ightarrow+\infty\)).

Answer:

The factored form of the polynomial \(f(x)=x^{4}+4x^{3}-28x^{2}-64x + 192\) is \(f(x)=(x - 2)(x + 6)(x - 4)(x + 4)\) and the graph can be plotted using the x - intercepts \((2,0)\), \((-6,0)\), \((4,0)\), \((-4,0)\), y - intercept \((0,192)\) and other calculated points.