so i am in precalculus, we need to graph the fuction using derivatives, maxima, intercepts and concavity. I am having trouble factoring cubics: for example: x^4-4x^3+16x --> x(x^3-4x^2+16)Now what? That is without calculator, how do you find the roots?
The equation x^3-4x^2+16=0 has 1 real root
x=-1.68 (approx)
(x^4-4x^3+16x)'=4x^3-12x^2+16=4(x^3-3x^2+4)=4(x+1)(x^2-4x+4)=4(x+1)(x-2)^2
(x^4-4x^3+16x)''=12x^2-24x=12x(x-2)
To find the roots of a cubic equation, you follow a series of steps. Here's how you can proceed with the given cubic equation: x^3 - 4x^2 + 16x.
Step 1: Factor out the common factor, if any.
In this case, you have already factored out the common factor, which is x.
Step 2: Examine the remaining expression inside the parentheses.
You have (x^3 - 4x^2 + 16x) left.
Step 3: Use various factorization techniques to factor the remaining expression.
Unfortunately, in many cases, factoring cubic polynomials without any given roots or known factors is quite complex. However, there are a few methods you can try.
Method 1: Rational Root Theorem
One possible approach is to use the Rational Root Theorem. The theorem states that if a rational number (p/q) is a root of the polynomial equation, then p must be a factor of the constant term (16 in this case), and q must be a factor of the leading coefficient (1 in this case). By testing various possible rational roots using synthetic division, you may be able to find actual roots of the equation.
Method 2: Synthetic Division
Another method, which may work in simpler cases, is synthetic division. However, synthetic division can only be used to find rational roots. You would have to test possible rational roots by substituting them into the synthetic division formula and checking if the remainder is zero for any of the roots.
Method 3: Factoring by Grouping or Recognizing Patterns
Occasionally, you may be able to recognize patterns or use factoring by grouping to factorize the cubic equation. However, these methods often involve identifying patterns or rearranging the terms in a specific way.
Step 4: Consider the possibility of using numerical methods.
If all the previous methods fail to find rational roots, you can resort to numerical methods like Newton's method or using a calculator, as you mentioned.
Keep in mind that factoring cubic polynomials can sometimes be challenging, especially without any given roots or known factors. If you are allowed to use a calculator, using numerical methods may be the easiest and most accurate way to find the roots of cubic equations.