help with q9

[visual aspen]

You’re given the equation of the polynomial, and two roots, so you can find the third root by writing it in terms of its factors, or by making an educated guess and seeing if it checks out

Since we know that (-3,0) is one of its roots, we also know that (x+3) is a factor of that polynomial

I don’t know which method of division you prefer to use, but once you’ve divided the equation by (x+3) you get (x^2-3x+2) which you can then factorise as a quadratic

That gives you the factors as (x+3)(x-1)(x-2)

This means that C has coordinates (2,0)

b)

It’s integration with respect to x

integrating x^3 - 7x + 6, in simple words we raise the power of x by 1 and divide by new power to get x^4/4 - 7x^2/2 + 6x + C

c)

Definite integration if I got the name correct

∫-3 to 2(x^3 - 7x + 6)

We can do this in two steps, one of them calculating the area under the graph for x values -3 to 1 and the other for x values 1 to 2

Integrate to get what we did in part b), and input the upper x value, then input the lower x value and subtract

For the x values from -3 to 1, we plug in x as 1 and then x as -3 into (x^4/4 - 7x^2/2 + 6x)

And then for the x values from 1 to 2 we plug in x as 2 and x as 1 into that same thing

The area that is below the x axis will likely get you a negative area

We add those up to get the total area of the area under the graph