Graphing a polynomial

[lriosmo]

I'm struggling with this problem: x[sup:tsdbwf4q]4[/sup:tsdbwf4q] - 5x[sup:tsdbwf4q]2[/sup:tsdbwf4q] + 4. I need to factor, find all intercepts, and graph. I think my problem is in the factoring. Please help! Thank you.

 

[Aladdin]

I think my problem is in the factoring. .

Factoring ax 4 + bx 2 + c
We can factor some trinomials of degree 4,for example, we factor x 4 +6x 2 + 5 as (x^2)^2 +6(x^2) + 5 : x 4 +6x 2 +5 =
(x^2)2 +6(x^2) + 5 = (x^2 +5)(x^2 + 1).

Many smiles

 

[mmm4444bot]

?
This polynomial is quadratic, in form. So, we could make a temporary substitution and then use the Quadratic Formula to find the roots.

For example: let z = x^2

Then we have z^2 - 5z + 4. Find the roots for this polynomial, and then set them equal to x^2 and solve for x.

Another approach is using the Rational Roots Theorem. That provides six possiblities, and it's a simple matter to check them.

Once we have the roots, the factorization is trivial.

What methods have you studied, so far? I mean, if you make some statements about what you already know or tried, I could provide specific help.

 

[lriosmo]

Thank you both. Using the first method I was able to solve for x. I got x=4 and x=1. I also found y-intercept = (0,4). Given a positive leading coefficient with an even exponent, I believe this results in a graph that rises at both the left and right. I hope this is correct! Thanks again.

 

[garf]

The polynomial you have has as a higher power 4, so it has 4 roots, yours have two positive and two negativ. The roots are:
x1= 1
x2 = -1
x3 = 2
x4 = -2
The number of roots of a polynomyal depends on the higher power of it, namely n, it has n roots ( not necesarily all of them real)

I hope this helps,
garf

 

[mmm4444bot]

Using the first method I was able to solve for x. I got x=4 and x=1.

I think that you mean, "I got z = 4 or z = 1".

Now, you need to reverse the previous substitution, to go back to x.

z = 4 means x^2 = 4.

z = 1 means x^2 = 1.

Solve these two equations for x, and you'll have a total of 4 roots.

You know that a root of x = 2 means that (x - 2) is a factor, yes? So, there are four factors in the requested factorization.

 

[mmm4444bot]

The number of roots of a polynomyal [of degree n] has n roots (not necesarily all of them real)

And, I will add, not necessarily distinct (see: "multiplicity").