Question

how do you graph h(x)=x^3+2x^2-x-2.

Answer 1

The polynomial function h (x ) = x ³ + 2x ²- x - 2

1) Zeros h (x ) = 3x ²(x - 1)(x - 2) using zeros to graph the polynomial.

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution.

The Rational Zero Theorem can be used for finding the some possible zeros to test

Possible values p/q  = ± 1,   ± 2.

Make a table for the synthetic division and test possible  zeros.

p/q	1	2	-1	-2
1	1	3	2	0

Since f(1) = 0,   x  = 1 is a zero. The depressed polynomial is   x ²- 3x + 2 = 0.

Since the depressed polynomial of this zero, x²- 2x + 5, is quadratic, now factorise to find the roots of the related quadratic equation.

x ²- 3x + 2 = 0

x ²- 2x - x + 2 = 0

x (x - 2) -1(x - 2) = 0

(x -1)(x - 2) = 0

Apply zero product property.

x - 1 = 0 and x - 2 = 0

x = 1 and x = 2

Rational zeros of x ³ + 2x ²- x - 2 are 1 , -1 , -2.

Real zeros are x  intercepts of the graph.

2)Test points

Make the table of values to for the polnomial.

Here i test 3 points to determine whether the graph of polynomials lies above or below the x axis.

Choose values for x and find the corresponding values for y.

x	y = x 3 + 2x 2- x - 2	(x, y )
- 0.5	y = (-0.5)3+2( -0.5)2 -(-0.5) - 2 = -1.125	(- 0.5, -1.125)
0	y = (0)3+2( 0)2 -(0) - 2 = -2	(0, -2)
0.5	y = (0.5)3+2( 0.5)2 -(0.5) - 2 = -1.875	(0.5, -1.875)

3) End behavior h (x ) = x ³ + 2x ²- x - 2

Degree of the polynomial is 3 and leading coefficient 1.

The graph of a polynomial function is always a smooth curve; that is, it has no breaks or corners.

All odd degree polynomials behave on thier ends like cubics.

All odd degree polynomials  have ends that head off in opposite directions.depending on whether the polynomial has, respectively, a positive or negative leading coefficient.

The above polynomial odd degree  polynomial with a positive leading coefficient .

So the graph falls to the left and rises to the right.

4)Graph

1.Draw a coordinate plane.

2.Plot the x  intercepts coordinate points found in the table.

3.Then sketch the graph, connecting the points with a smooth curve.