f(x) =x^4 - x^3 -4x^2 +4. find factor and graph.

I only factor to this step:
(x^3 - 4x-4)(x-1) = f(x)
but i don't know how to continue this step to require factor more completely and graph it. Please help. this is algebra 2. Thank you.

http://www.mathportal.org/calculators/polynomials-solvers/polynomial-graphing-calculator.php

note
two real roots
1
2.383
two complex roots

f(x) =x^4 - x^3 -4x^2 +4. find factor and graph.

I only factor to this step:
(x^3 - 4x-4)(x-1) = f(x)
but i don't know how to continue this step to require factor more completely and graph it. Please help. this is algebra 2. Thank you.
this graph cannot allow to use graphic calculator, must be use sketch graph by hand only. Please help. Thank you.

To fully factor the polynomial f(x) = x^4 - x^3 - 4x^2 + 4, we can use a combination of factoring techniques and the Rational Root Theorem. Here's how you can do it:

Step 1: Look for rational roots using the Rational Root Theorem.
The rational root theorem states that if a polynomial has a rational root (p/q), then p must be a factor of the constant term (in this case, 4), and q must be a factor of the leading coefficient (in this case, 1). The possible rational roots are therefore the divisors/factors of 4, which are ±1, ±2, and ±4.

Step 2: Test the possible rational roots using synthetic division.
Using synthetic division, you can test each of the possible rational roots to see if they are roots of the polynomial. Begin by dividing the polynomial by (x - p), where p is a possible root. For our given polynomial, we can start with (x - 1) because it's the simplest factor found in your work.

1 | 1 -1 -4 4
|_____________
1 0 -4 0

Since the remainder is 0, it means (x - 1) is a factor of the polynomial. The quotient obtained from the division is the reduced polynomial.

Step 3: Factor the reduced polynomial further.
The reduced polynomial obtained after division is x^3 - 4x - 4. To factor it further, you can use various techniques like factoring by grouping, factoring special forms such as a difference of cubes or sum of cubes, or using trial and error. However, in this case, factoring by grouping seems like the best approach.

x^3 - 4x - 4 can be rewritten as:
(x^3 - 4x) - 4
x(x^2 - 4) - 4
x(x + 2)(x - 2) - 4

So the fully factored form of f(x) = x^4 - x^3 - 4x^2 + 4 is:
f(x) = (x - 1)(x + 2)(x - 2)(x^2 + 4)

Step 4: Graph the function.
To graph the function, plot the x-intercepts (the points where the function crosses the x-axis) and identify any additional key points.

In this case, the x-intercepts are x = 1, x = -2, and x = 2.

Additionally, since the degree of the polynomial is 4, it suggests the possibility of a point of inflection or local maximum/minimum. However, to find these key points, we need to find the derivative of f(x) and solve for the critical points.

Taking the derivative of f(x) = x^4 - x^3 - 4x^2 + 4, we get:
f'(x) = 4x^3 - 3x^2 - 8x

Setting the derivative equal to zero, we can solve for critical points:
4x^3 - 3x^2 - 8x = 0

Solving this cubic equation might involve factoring, using synthetic division, or using numerical methods (such as graphing calculators or numerical solvers) to find the critical points. Once you find the x-values of the critical points, you can evaluate f(x) at those points to find the corresponding y-values.

Finally, plot the x-intercepts and any additional key points you found on the graph paper to sketch the graph of f(x).