Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
`f(x) = x^3 - 2x^2 - 13x - 10`
Find the real zeros of f. Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
b. There are no real zeros
Since `f(x) = x^3-2x^2-13x-10`
we can evaluate the function at all factors of `pm 10` to find possible zeros of the function. In this case, we see that
which means that `x+1` is a factor of `f(x)` . Now use division to get the remaining factors, which gives:
But the quadratic factor will further factor into two linear terms to get
This means that the zeros of the cubic are the negatives of the linear factors to get `x=-1` , `x=-2` and `x=5` .
Graphicof cubic show the roots: `x_1=-1;x_2=5;x_3=-2`
`f(-1)=0` so that: `x_1=-1`
so that we can write:
So: concerning solutions : `x^2-3x-10` :
`Delta= 9-4(-10)=49 >0` two real solution:
`x=(3+-sqrt(49))/2=(3+-7)/2` `x_2=5` `x_3=-2`
Finally the solution given equation are: