the following system of equations can be used to find the roots of the equation x^3 + 72 = 5x^2 + 18x. y =…

the following system of equations can be used to find the roots of the equation x^3 + 72 = 5x^2 + 18x. y = x^3 + 72 y = 5x^2 + 18x graph this system of equations on the graphing calculator. how many intersection points can you see in the default viewing window?

Answer

Explanation:

Step1: Set the two equations equal

Since at intersection points $y$ - values are equal, set $x^{3}+72 = 5x^{2}+18x$.

Step2: Rearrange to standard polynomial form

$x^{3}-5x^{2}-18x + 72=0$.

Step3: Try to find roots by trial - and - error

We test some simple values of $x$ like $\pm1,\pm2,\pm3,\pm4,\pm6,\pm 8,\pm9,\pm12,\pm18,\pm24,\pm36,\pm72$. When $x = 3$, $3^{3}-5\times3^{2}-18\times3 + 72=27-45 - 54+72=0$. So $(x - 3)$ is a factor.

Step4: Use polynomial long - division or synthetic division

Dividing $x^{3}-5x^{2}-18x + 72$ by $(x - 3)$ gives $x^{2}-2x - 24$.

Step5: Factor the quadratic

$x^{2}-2x - 24=(x - 6)(x + 4)$.

Step6: Find all roots

Set each factor equal to zero: $x-3=0$ gives $x = 3$, $x - 6=0$ gives $x = 6$, $x+4=0$ gives $x=-4$. There are 3 roots.