Someone me i dont understand
one factor of the function f(x) = x3 − 6x2 + 11x − 6 is (x − 3). describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technology. include all intercepts in your answer.

x-inrercept: (1,0)

y-intercept: (0,-6)

Step-by-step explanation:

To find x-inrercepts, you need to set the function equal to 0 [aka, set f(x)=y=0] and solve for x. To find y-intercepts, you need to plug 0 into the function itself [aka, set x=0].

The y-intercept is simple here. Plugging 0 into the function wipes everything out except for the -6. So your point is (0,-6).

The x-intercept is... a lot more involved to figure out. You're given the hint of (x-3) being one of the factors, but personally I didn't use that. I'll try to show the steps I took below, but what I ended up doing was factoring as much as I could until I hit a wall. And then I eyeballed what values of x would give me an answer. It looked a little like this:

f(x)= x³-6x²+11x-6

=x(x²-6x+11)-6

=x(x²-6x+9-9+11)-6

=x(x²-6x+9-2)-6

=x[(x-3)²-2]-6

When I set the last one equal to 0, I found that 1 and 3 both worked out. Lastly, I plugged 1 and 3 into the original polynomial and found that only 1 gave me an output of 0. So the only x-intercept is (1,0).

I aslo double checked these points using technology, so they should be correct.

Edit: Aw man, the other person used long division! Can't believe I didn't think to do that! I'd definitely say that is the best method. Especially since I missed 2 as a factor...

Check the picture below.

so when we divide a polynomial by one of its roots, the remainder is always 0, and the quotient is simply another factor, so after the division of these two, we have that

now, to find the x-intercepts, recall all we do is, set y = 0 and then solve for "x", and likewise to find the y-intercepts, we set x = 0 and solve for "y".

step-by-step explanation:

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