sketch y=(x+2)(x-1)(x-4)

Question

solve the polynomial y=(x+2) (x-1) (x-4).

Answer 1

The polynomial function y = (x + 2) (x - 1)(x - 4)

1) Zeros y = (x + 2) (x - 1)(x - 4) using zeros to graph the polynomial.

Real zeros are x  intercepts of the graph.

To find x intercepts substitute y = 0 in the function.

(x + 2) (x - 1) (x - 4) = 0

Apply zero product property.

x + 2 = 0 , x - 1 = 0 and x  - 4 = 0

x = -2 , x = 1 and  x = 4.

x intercepts are (0,-2), (0,1) and (0,4).

2)Test points

Make the table of values for the polnomial.

Here i test 4 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 + 2) (x - 1)(x - 4)	(x, y )
-1	y = (-1 + 2) (-1 - 1)(-1 - 4) = -10	(- 1, 10)
0	y  = (0 + 2) (0 - 1)(0 - 4) = 8	(0, 8)
1	y = (1 + 2) (1 - 1)(1 - 4) = 0	(1, 0)
2	y = (2 + 2) (2 - 1)(2 - 4) = -8	(2, -8)

3) End behavior y = (x + 2) (x - 1)(x - 4)

y =  x ³ - 3x ²- 6x + 8

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 and coordinate points found in the table.

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

Solution For y = 0 then x = -2 , 1 , 4.

For x = 0 then  y = 8.