Rational Zeros of a Function

Normally zeros of a function mean when we plug the values for the variables the functions values tends to be zero.  Let us consider if we are having a function with variable x and we have a set of solution p(x) if we plug the solution for the given variables present in the function we will get the function f(x) = 0. To find the rational zeros we have to use the rational zeros theorem.

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Rational Zeros of a Function – Examples:

The rational theorem states that if we are having the polynomial p(x) wit the integer coefficients and we are having the zeros of the polynomial `p / q` then we can say `p(p / q) = 0` . Here p is nothing but the constant term of the polynomial and the q I nothing but the leading coefficient of the polynomial p(x). We will see some examples for finding the rational zeros of a function.

Example 1 for rational zeros of a function:

Find the rational zeros of the following function.

2x2 + 12x + 10

Solution:

The given function is

2x2 + 12x + 10

First we have to find the rots of the constant term. ±1, ±2, ±5, ±10

Now the leading co – efficient of the constant term is 2. So we have to divide by 2.

±`p / q` = ±`1/ 2` , ±`2/ 2` , ±`5/ 2` , ±`10/ 2`

= ±`1/2` , ±1, ±`5/2` , ±5

Now we have to use the synthetic division method to find the rational zeros.

1 / 2 | 2           12        10

|               1         13/2

|_____________________

2           13        33/2      = not a zero

-1 / 2 | 2           12        10

|               -1         -11/2

|_____________________

2           11        9/2      = not a zero



1 | 2           12        10

|               2         24

|_____________________

2           24        34      = not a zero

-1 | 2           12        10

|               -2        -10

|_____________________

2           10        0      = is a zero

5/2 | 2           12        10

|               5        85/2

|_____________________

2           17        105/2      = not a zero

- 5/2 | 2           12        10

|               -5        -35/2

|_____________________

2           7        15/2      = not a zero

5 | 2           12        10

|              10        110

|_____________________

2           22        120      = not a zero

-5 | 2           12        10

|              -10      -10

|_____________________

2           2        0      = is a zero

So from the above the rational zeros of the functions are p(x) is -1 and -5

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Rational Zeros of a Function – more Examples:

Example 2 for rational zeros of a function:

Find the rational zeros of the following function.

x2 + 4x + 3

Solution:

The given function is

x2 + 4x + 3

First we have to find the rots of the constant term. ±1, ±3

Now the leading co – efficient of the constant term is 1. So we have to divide by 1.

±`p / q` = ±`1/ 1` , ±`3 / 1`

= ±1, ±3

Now we have to use the synthetic division method to find the rational zeros.

1 | 1           4        3

|               1       5

|_____________________

1           5       8      = not a zero

-1| 1           4        3

|              -1      -3

|_____________________

1            3        0      = is a zero

3 | 1           4        3

|              3        21

|_____________________

1          7       24      = not a zero

-3| 1           4        3

|             -3      -3

|_____________________

1            1        0      = is a zero

So from the above the rational zeros of the functions are p(x) is -1 and -3

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