Question

What are the standard form and the factored form of the function?

Use the zeros to find all of the linear factors of the polynomial function.

Write the equation of the graphed function f(x), where a is the leading coefficient. Use the factors found in part D. Express the function as the product of its leading coefficient and the expanded form of the equation in standard form.

Use the y-intercept of the graph and your equation from part E to calculate the value of a.

Given what you found in all of the previous parts, write the equation for the function shown in the graph.

Answer 1

f(x)= -(x^3+10x^2-275x-1500) Standard

f(x)= -(x+20)(x+5)(x-15)

Explanation:

The factored form of the equation is:

(x+20)(x+5)(x-15)=f(x) Each of the zeroes are where the graph crosses the x-axis.

The expanded form of the equation is found by using FOIL on the equations above:

-x^3+10x^2-275X-1500

The leading coefficient is negative because the graph rises to the left and falls to the right.

Answer 2

Factored form:
`P(x)=-(1)/(1500)(x+20)(x+5)(x-15)`.

Standard form:
`P(x)=-(x^3)/(1500)-(x^2)/(150)+(11x)/(60)+1`

Explanation:

The factor form of a polynomial is

`P(x)=a(x-c_1)(x-c_2)...(x-c_n)`

where, a is a constant and leading coefficient,
`c_1,c_2, ...,c_n` are n zeroes of the polynomial.

From the given graph it is clear that the x-intercepts of the function are -20, -5 and 15. It means the zeroes of the given function are -20, -5 and 15. So, the required function is

`P(x)=a(x-(-20))(x-(-5))(x-15)`

`P(x)=a(x+20)(x+5)(x-15)` .... (1)

From the given graph it is clear that y-intercept of the function is (0,1). Use the y-intercept of the graph to find the value of a.

`1=a(0+20)(0+5)(0-15)`

`1=-1500a`

Divide both sides by -1500.

`(1)/(-1500)=a`

`-(1)/(1500)=a`

Put
`a=-(1)/(1500)` in equation (1).

`P(x)=-(1)/(1500)(x+20)(x+5)(x-15)`

Therefore the factored form of the function is
`P(x)=-(1)/(1500)(x+20)(x+5)(x-15)`.

Expand the above function to find the standard form of the function

`P(x) = (-x^3 - 10 x^2 + 275 x + 1500)/(1500)`

`P(x)=-(x^3)/(1500)-(x^2)/(150)+(11x)/(60)+1`

Therefore the standard form of the function is
`P(x)=-(x^3)/(1500)-(x^2)/(150)+(11x)/(60)+1`.