For the graph f(x)=x^3+2x^2+x determine the x along with their multiplicities

Answer 1

To find the x-intercepts of the graph of f(x), we need to solve the equation f(x) = 0.

f(x) = x^3 + 2x^2 + x

Setting f(x) = 0:

x^3 + 2x^2 + x = 0

Factoring out an x from each term:

x(x^2 + 2x + 1) = 0

The quadratic equation x^2 + 2x + 1 can be factored:

(x + 1)(x + 1) = 0

(x + 1)^2 = 0

To solve for x, we set each factor equal to zero:

x + 1 = 0

x = -1

So the x-intercept of the graph is -1. The multiplicity of the x-intercept is 2, since (x + 1) is squared in the factored form.