# Calculus of Polynomials

We can use the graphing techniques to sketch the graph of a polynomial.

Find the zeroes of a polynomial or the roots of the polynomial equation to sketch its graph.

For example, sketch the graph of the polynomial $P(x)={ x }^{ 3 }+{ x }^{ 2 }-6x$

First, we write the polynomial P(x) as a product of its factors:

$P(x)={ x }^{ 3 }+{ x }^{ 2 }-6x\\ \qquad =x\left( { x }^{ 2 }+x-6 \right) \\ \qquad =x(x+3)(x-2)$

Then we find the x-intercepts: y=0

$0=x(x+3)(x-2)\\ x=0,\quad x+3=0,\quad x-2=0\\ x=0\quad OR\quad x=-3\quad OR\quad x=2$

So the x-intercepts are 0, -3 and 2.

For y-intercepts: x=0

$P(0)={ (0) }^{ 3 }+{ (0) }^{ 2 }-6(0)=0$

So the y-intercept is 0.

We look at which part of the graphs are above and which are below the x-axis between the x-intercepts.

Test 1) x=-4 (less than -3)

$P(-4)=-4(-4+3)(-4-2)\\ \qquad =-24\qquad <\qquad 0$

So the curve is below the x-axis.

Test 2) x=-1 (between -3 and 0)

$P(-1)=-1(-1+3)(-1-2)\\ \qquad =6\qquad >\qquad 0$

So the curve is above the x-axis.

Test 3) x=1 (between 0 and 2)

$P(1)=1(1+3)(1-2)\\ \qquad =-4\qquad <\qquad 0$

So the curve is below the x-axis.

Test 4) x=3 (more than 2)

$P(3)=3(3+3)(3-2)\\ \qquad =18\qquad >\qquad 0$

So the curve is below the x-axis.

Then by the information above, we can sketch the graph: