How to solve x^3 +5x^2 +4x-20?

Question

'm supposed to find the x and y intercepts the y intercept is -20 i know but how do i find the x intercept, if not by solving. if the only way to find it is to solve it then please show me how.

f(x)=x^3 +5x^2 +4x-20

Answer 1

The function y  = x ³ + 5x ² + 4x  - 20

Find the y  - intercept by substituting x  = 0 in the given equation.

y  = 0³ + 5(0)² + 4(0) - 20

y  = 0 + 0 + 0 - 20

y  = - 20

The y  - intercept is - 20.

Find the x - intercept by substituting y  = 0 in the given equation.

x ³ + 5x ² + 4x - 20 = 0

Graph the above polynomial function to find x  intercepts.

Test points

Make the table of values for the polynomial.

Here i test 5 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 = x3 + 5x2 + 4x - 20	(x, y )
-1	y = (-1)3 + 5(-1)2 + 4(-1) - 20 = -20	(- 1, -20)
-2	y = (-2)3 + 5(-2)2 + 4(-2) - 20 = -16	(-2, -16)
1	y = (1)3 + 5(1)2 + 4(1) - 20 = -10	(1, -10)
2	y = (2)3 + 5(2)2 + 4(2) - 20 = 16	(2, 16)
-4	y = (-4)3 + 5(-4)2 + 4(-4) - 20 = -20	(-4, -20)

End behavior y  = x ³ + 5x ² + 4x  - 20

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 their 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.

1.Draw a coordinate plane.

2.Plot the y  intercepts and coordinate points found in the table.

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

Graph :

The points where it crosses the x  axis  will give solutions to the polynimial function .

The graph crosses the x  - axis at a point that would suggest a factor.

It crosses the x  - axis at one point hence there are one real root.

x  = 1.48

Answer 2

Contd...

Use synthetic division to determine if the given value of x  is a root of the polynomial.

Since f (1.48) = 0, x = 1.48 is a zero.

The depressed polynomial is

Since the depressed polynomial of this zero, , is quadratic,

Use the Quadratic Formula to find the roots of the related quadratic equation

 

Roots are

The cubic function has two imaginary solutions and one real solution.

Therfore, x - intercept is 1.48.