Factoring f(x)=x^3+x to Find x-Axis Intersection

In summary, the conversation discusses using factors to show that a function with the given equation of f(x)=x^3+x crosses the x-axis once only. The participant factors the function to x(x^2+1) and realizes that the x^2+1 factor has no real zeros, but the x factor has one real zero at x=0. This fulfills the property of an x-intercept for the graph. The participant finds understanding in this and the summary concludes with a confirmation of their newfound understanding.
  • #1
david18
49
0
Ive got a function of f(x)=x^3+x and I need to use factors to show that the graph crosses the x-axis once only.

I just factorised it to x(x^2+1) which isn't very helpful, and if i divide everything by x and complete the square with x^2+1 i get a negative number and stuff...

any help?
 
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  • #2
david18 said:
Ive got a function of f(x)=x^3+x and I need to use factors to show that the graph crosses the x-axis once only.

I just factorised it to x(x^2+1) which isn't very helpful,
Oh yes it is!

So, you have found that f(x)=x(x^2+1)

What property, i.e, which equation must an x-intercept of the graph fulfill?
 
  • #3
x(x^2+1)=0
 
  • #4
david18 said:
Ive got a function of f(x)=x^3+x and I need to use factors to show that the graph crosses the x-axis once only.

I just factorised it to x(x^2+1) which isn't very helpful, and if i divide everything by x and complete the square with x^2+1 i get a negative number and stuff...

any help?

Good is that you factored the function. the x^2 + 1 factor has no "real" zeros; but the x factor has one real zero, being 0. (zero).

Let me try again in case this helps.
(x^2 + 1) = 0 for what real values of x? For NONE. We do not usually graph complex zeros in two dimensional cartesian plane (at least for our purposes here).

When is x=0 (using the other factor)? when x=0, already shown.
 
  • #5
hi, thanks for the reply, it all makes sense now :)
 

FAQ: Factoring f(x)=x^3+x to Find x-Axis Intersection

How do I factor a function to find the x-axis intersection?

To factor a function, you need to find the roots of the function, which are the values of x where the function intersects with the x-axis. To do this, you can set the function equal to 0 and solve for x using factoring techniques. In this case, the function is f(x)=x^3+x, so setting it equal to 0 gives you x^3+x=0. By factoring out an x, you get x(x^2+1)=0. The roots are x=0 and x=±i, where i is the imaginary unit.

Why is it important to find the x-axis intersection of a function?

The x-axis intersection of a function is important because it gives you the x-values at which the function crosses or touches the x-axis. These values can provide information about the behavior of the function, such as its zeros, turning points, and end behavior. They can also help you graph the function and solve problems related to the function.

Can I use the quadratic formula to factor this function?

No, you cannot use the quadratic formula to factor this function because it is a cubic function, meaning it has a degree of 3. The quadratic formula is used to solve quadratic equations, which have a degree of 2. To factor a cubic function, you need to use other factoring techniques, such as grouping or the rational root theorem.

Are there any other methods to find the x-axis intersection besides factoring?

Yes, there are other methods to find the x-axis intersection of a function. One method is to use a graphing calculator or software to plot the function and visually determine the x-values where it crosses the x-axis. Another method is to use the bisection method, where you test different values of x to see when the function equals 0. However, factoring is often the most efficient and accurate method to find the x-axis intersection.

What if the function cannot be factored to find the x-axis intersection?

If the function cannot be factored, it means that it has no real roots, and therefore does not intersect the x-axis. In this case, the function may have complex roots or no roots at all. You can still graph the function and use other methods, such as the ones mentioned in question 4, to approximate the x-axis intersection. Keep in mind that some functions may not have an x-axis intersection at all, such as those with a vertical asymptote instead.

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