Hi guys . I m  Tajinderpreet i m going to write about rational root theorem .

RATIONAL ROOT THEOREM

The rational root theorem, or Rational Root Test can be used to do a quick check whether a polynomial has rational roots.

A root or zero of a function is a number that, when plugged in for the variable, makes the function equal to zero. Thus, the roots of a polynomial P(x) are values of x such that P(x) = 0 .

Steps :-

1. Arrange the polynomial in descending order.
1. Write down all the factors of the constant term. These are all the possible values of p .
2. Write down all the factors of the leading coefficient. These are all the possible values of q .
3. Write down all the possible values of . Remember that since factors can be negative, and - must both be included. Simplify each value and cross out any duplicates.
4. Use synthetic division to determine the values of for which P() = 0 . These are all the rational roots of P(x) .

Here is some examples :-

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Example: Find all the rational zeros of P(x) = x ³ -9x + 9 + 2x ⁴ -19x ² .


P1. P(x) = 2x ⁴ + x ³ -19x ² - 9x + 9

Factors of constant term: ±1 , ±3 , ±9 .

Factors of leading coefficient: ±1 , ±2 .

possible values of p ; +1 + 1 +3 + 3 +9 +9 .

q 1 2 1 2 1 2

Use synthetic division:

divide by 1 ) 2 1 -19 -9    9

                       2   3   -16 -25

                    2 3 -16 -25 -16

Remainder = -16 . not zero

divide by -1) 2 1 -19 -9  9

                        -2   1  18 -9

                     2 -1 -18  9  0

Remainder =0 . is zero

 

divide by 1/2)  2 1 -19 -9   9

                           1    1   -9  -9

                        2 2  -18  9   0

Remainder = 0 . is zero

 divide by -1/2)  2 1 -19  -9       9

                              1    0   19/2 -1/4

                           2 0 -19 1/2    35/4

Remainder = 35/4 . not zero

divide by 3)  2 1 -19 -9  9

                        6  21  6 -9

                    2  7  2   -3 0

Remainder =0 .is zero

Thus, the rational roots of P(x) are x = - 3 , -1 , , and 3 .

We can often use the rational zeros theorem to factor a polynomial. Using synthetic division, we can find one real root a and we can find the quotient when P(x) is divided by x - a . Next, we can use synthetic division to find one factor of the quotient. We can continue this process until the polynomial has been completely factored.

Example : 2

Consider: x²-3x+2=0

If this has rational roots p/q (which we know it has), then p is a factor of 2 and q is a factor of 1.

So the roots are some combination of the factors in the fraction below:

We must use all the numbers, that is we need to use both 1 and 2 to make possible roots. Where all the roots are rational, we cannot use just the "1" as a numerator. We need to use the 2 also. So 1 and 1 won't do as numerators, although 2 and 2 are possible (because the 1 is always automatically included in any integer.)

We test each of the roots in the original equation and, in this case, we find x=1/1 or 2/1, which are 1 and 2, of course.

  

HOPE this will help you :)