Question

list all possible factors for h(x)=4x^4-5x^3+2x^2-x+5. the way I answered was I wrote down the possible roots, domain/range, the asymptotes, and x and y intercepts. would this be correct?

Answer 1

Factors are usually written as (x -#) or (x+#). You can use the x intercepts to find the REAL factors. If there are imaginary factors you will have to find them by solving using quadratic formula after you find the other 2 factors.

Answer 2

the way I got the x intercepts though there are none? so what would I do from here?

Answer 3

Have you learned the rational root theorem?

Answer 4

You listed all of the possible factors by using the rational root theorem, correct?

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient.

i.e.,
\[\pm\frac{ 1 }{ 2 }, \pm \frac{ 1 }{ 4 }, \pm1, \pm \frac{ 5 }{ 2 }, \pm \frac{ 5 }{ 4 }, \pm5\]

If that is the case you are done. These are possible factors. Note that there are no REAL factors for the equation you provided.

Answer 5

i don't really understand it

Answer 6

The problem asks for all possible factors, and that's exactly what the RRT provides.

Answer 7

so is there no real factors that i need to put down on my assignment then?

Answer 8

When you multiply polynomial s, the leading and trailing coefficients in the answer are formed by multiplying the leading and trailing coefficients of the factors. This means you can guess what they must be, if they exist, by working backwards, looking only at the leading and trailing coefficients. It is MUCH easier to see if the leading coefficient is 1...

Answer 9

No, you need to work out the whole set of possible factors, even though it turns out none of them are factors.

Answer 10

Simple example:
\[x^2+4x+3\]
Any possible factors are going to be of the form \((x\pm1)\) or \((x\pm3)\) because the leading coefficient is 1 and the factors of 3 are 1 and 3

Answer 11

We don't necessarily know if they are positive or negative in all cases because a negative number to an even power is positive, and of course so is a positive number.

Answer 12

so because the trailing coefficient is 5 its factors are \[(x \pm1) and (x \pm5)\]

Answer 13

Typing math on iPad is too painful, so I'm not going to do a more complicated example where leading coefficient is other than 1

Answer 14

See, first, to find out the roots of the function, check the intervals in which it is increasing and decreasing. In short, check the monotonicity of the function. And also check the concavity changes. Study the graph, it's behaviour.

So, if h"(x) is positive throughout the domain, we can infer that the graph remains concave upwards always. So, h(x) is some parabola sort of a shape.
Check for h(0). h(0)=5.