Math  /  Algebra

QuestionWhat are the xx-intercepts of the graph of f(x)=x33x24xx2+6x+5f(x)=\frac{x^{3}-3 x^{2}-4 x}{x^{2}+6 x+5} ? 1,5-1,-5 0,4 1, 5 0,4,10,4,-1

Studdy Solution

STEP 1

What is this asking? Where does this graph cross the x-axis? Watch out! Don't forget: an x-intercept happens when y=0y=0, but we also need to watch out for values of xx that make the denominator zero!

STEP 2

1. Factor the numerator
2. Factor the denominator
3. Simplify the function
4. Find the x-intercepts

STEP 3

Alright, let's **factor** that numerator!
We've got x33x24xx^3 - 3x^2 - 4x.
First, we can see that each term has an xx, so let's **factor** that out: x(x23x4)x(x^2 - 3x - 4).

STEP 4

Now, let's see if we can **factor** the quadratic.
We're looking for two numbers that multiply to 4-4 and add up to 3-3.
Those numbers are 4-4 and 11!
So, we can rewrite the numerator as x(x4)(x+1)x(x-4)(x+1).
Awesome!

STEP 5

Now, let's **factor** the denominator, x2+6x+5x^2 + 6x + 5.
We need two numbers that multiply to 55 and add up to 66.
That's 55 and 11!
So, the factored denominator is (x+5)(x+1)(x+5)(x+1).

STEP 6

Now, we can put the factored numerator and denominator together: f(x)=x(x4)(x+1)(x+5)(x+1) f(x) = \frac{x(x-4)(x+1)}{(x+5)(x+1)} Notice that we have an (x+1)(x+1) in both the numerator and the denominator!
As long as xx isn't 1-1, we can divide the numerator and denominator by (x+1)(x+1), effectively turning the (x+1)(x+1) terms into ones, and simplifying our function: f(x)=x(x4)x+5 f(x) = \frac{x(x-4)}{x+5} Keep in mind that the function is *undefined* at x=1x=-1, so there can't be an intercept there.

STEP 7

Remember, the x-intercepts happen when f(x)=0f(x) = 0.
So, we set our simplified function equal to zero: x(x4)x+5=0 \frac{x(x-4)}{x+5} = 0

STEP 8

To solve this, we can multiply both sides by (x+5)(x+5).
As long as xx is not 5-5, we can multiply both sides by (x+5)(x+5) to get x(x4)=0x(x-4) = 0.

STEP 9

Now we have a product equal to zero, which means either x=0x=0 or x4=0x-4=0.
If x4=0x-4=0, then x=4x=4.
So, our x-intercepts are x=0x=0 and x=4x=4!

STEP 10

The x-intercepts are 00 and 44.

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