How to Find All the Zeros of a Function

When you find all the zeros of a polynomial function, you get the values of x which make the function f(x) equal to zero. A zero can be a real or complex number. At times, finding all the zeros of functions that have a degree of 3 or higher can be a tedious task. You should establish a plan to break down these higher degree polynomial functions into workable factors.

Instructions

Understand the Technique of Finding Zeros of a Function

1. Discover the total number of roots of a function by using the corollary to the fundamental theorem of algebra which states that any polynomial of degree n has exactly n total roots or zeros for a function. This does not tell us if these zeros are all real numbers.

2. Determine the maximum number of possible real roots by using Descartes' rule of signs. For a function f(x), count the number of sign changes for the x terms. This is the maximum number but may not be the actual number of possible positive real roots or zeros. To find all possibilities decrease this number by multiples of 2 until the result is negative. You determine the possible negative real roots by finding f (-x) and then determining the zeros in the same manner as above (see Resources).

3. Apply the rational root theorem which states that a polynomial with leading coefficient A and constant term C can have rational roots or zeros of the form ± p/q where p is a factor of C and q is a factor of A. It does not tell us which are the actual roots.

4. Use synthetic division to find which one of the possible rational roots is an actual root. You first choose a possible rational root ± p/q from the list in Step 3. Then use synthetic division (see Resources). If a root is complex then you need to find the root by the procedure in Step 5.

5. Find the remaining roots of quadratic equations which are not factorable by using the quadratic formula. For a quadratic equation in its standard form, ax²+bx+c=0, the formula is x= [-b ±sqrt (b²-4ac)]/2a. Note that you can use the quadratic equation to also find factorable quadratic equations.

Find all Zeros of a Function

1. Use the polynomial function f(x) =3x³-8x² +5x-2 as an example and use the technique outlined in Section 1 to find the zeros or roots. First look at the degree of the polynomial, it is 3 so there are exactly 3 zeros or roots for this function. They may be real or complex zeros.

2. Start at the left side of the equation for the example in Step 1 and use Descartes' rule of signs to count the number of sign changes for the x terms; there are 3 sign changes. Subtract 2 from this number (3-2) to get 1, so there are either 3 or 1 real positive roots or zeros for this function. Now find the negative real roots by determining f (-x). For the example in Step 1, f (-x) = 3(-x³)-8(-x²) +5(-x)-2= -3x³-8x² -5x-2. Here there are no sign changes so there are no negative real roots or zeros.

3. Apply the rational root theorem to the equation of the example 3x³-8x² +5x-2 and find ± p/q. First find the factors of the constant term 2 which are 1, 2. Then find the factors of the leading coefficient 3 which are 1 and 3. The possible rational roots ± p/q is ± 1, ± 2, ±1/3, and ±2/3.

4. Find an actual root of the example by first choosing a rational root from the list in Step 3 and then use the procedure for synthetic division (see Resources below). You get that 2 is the only rational root or zero and that (x-2) is a factor. Then the polynomial factors into (x-2) X (3x²-2x+1).

5. Set (x-2) X (3x²-2x+1) =0 and solve (x-2) =0 to get x=2. Then solve (3x²-2x+1) = 0.Use the quadratic formula from Section 1, Step 5 to find the last two zeros of the function because (3x²-2x+1) is not factorable. The complex pair [1+i(√2)]/3 and [1-i(√2)]/3 are the final two zeros of the function. The zeros of the polynomial function 3x³-8x² +5x-2 are x= 2, x= [1+i(√2)]/3 and x=[1-i(√2)]/3. This is the solution to this problem.

Resources

• Here is information on Descartes' rule and synthetic division.