Find the zeros of x 3 x
Calculate zeros of cubic functions
In general, a cubic function has the following form
\ (f (x) = ax ^ 3 + bx ^ 2 + cx + d \)
- Guess the zero
- Apply polynomial division
- Find the zero of the calculated term
\ (f (x) = 2x ^ 3 + 4x ^ 2 - 2x - 4 \)
Step 1: Guess the zero
Yes, you've read that correctly. You should guess a zero. Of course, this only works if the zero point is not too difficult to find. In school it is usually sufficient if you use the integer values between -3 and +3.
First guess: Zero at \ (x = 0 \)?
\ (f (0) = 2 \ times 0 ^ 3 + 4 \ times 0 ^ 2 - 2 \ times 0 - 4 = -4 \ neq 0 \)
Second guess: Zero at \ (x = 1 \)?
\ (f (1) = 2 \ times 1 ^ 3 + 4 \ times 1 ^ 2 - 2 \ times 1 - 4 = 0 \)
Excellent! We found a zero by guessing. Now we apply the polynomial division in order to find the other two zeros as quickly as possible.
Note: In the article "Solving Cubic Equations" we learn a simple procedure that helps us guess a zero.
2nd step: Apply polynomial division
The polynomial division proceeds in such a way that we divide our function by \ ((x-1) \). It is divided by \ ((x-1) \) because there is a zero at \ (x = 1 \). If the zero were at \ (x = -3 \), one would divide by \ ((x + 3) \).
\ [2x ^ 3 + 4x ^ 2 - 2x - 4: (x-1) = \ quad? \]
Note: In the article "Polynomial Division" you will find this example explained in detail!
End situation (after the polynomial division)
\ [2x ^ 3 + 4x ^ 2 - 2x - 4: (x-1) = 2x ^ 2 + 6x + 4 \]
By the way: The Horner scheme is a simple alternative to polynomial division!
3rd step: Find the zero of the calculated term
We get the other two zeros by solving the quadratic equation we calculated for the polynomial division.
\ (2x ^ 2 + 6x + 4 = 0 \)
This is the same equation that was discussed in the 2nd example in the section "Zeros of quadratic functions". The two zeros are called: \ (x_2 = -2 \) and \ (x_3 = -1 \). Since we have already guessed a zero - namely \ (x_1 = 1 \) - we have found all three zeros of this equation.
Zeros and their calculation
As Zero one denotes the x-coordinate of the intersection of a function graph with the x-axis. Since the y-coordinate of this intersection point is always zero, one can say: zeros are those x-values which, when the function starts, deliver the function value zero.
The zero of a linear function is obtained by setting the function equal to zero and then using equivalence transformations to solve for \ (x \).
The zeros of a quadratic function are usually calculated using the midnight formula. The pq formula or Vieta's theorem are also suitable for calculating the zeros of quadratic functions.
In order to calculate the zero of a cubic function, one must first guess a zero. Subsequently, one simplifies the term with the help of the polynomial division or the Horner scheme. In this way one again obtains a quadratic function which can be solved with the methods already mentioned above.
It is easiest if the functional term can be factored completely.
Then you can use the theorem of the zero product to calculate the zeros.
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