# Find the exact value of the zero between 4 and 5 using the equation P(x)=x^3- 7x^2+ 11x+ 3.

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Find the exact value of the root that lies between 4 and 5:

Given `x^3-7x^2+11x+3` from the rational root theorem the only possible rational roots are `+-1,+-3` . We find that 3 is a root so (x-3) is a factor:

`x^3-7x^2+11x+3=(x-3)(x^2-4x-1)`

To find the root between 4 and 5 we solve the quadratic factor:

`x^2-4x-1=0`

`x^2-4x=1`

`x^2-4x+4=1+4`

`(x-2)^2=5`

`x-2=+-sqrt(5)`

`x=2+-sqrt(5)`

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The root that lies between 4 and 5 is `2+sqrt(5)`

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The decimal expansion is only an approximation. Only the radical expression gives the exact value.

We can factor the polynomial as (x - 3)(x^2 - 4x -1).

Hence, x = 3 is a zero of our polynomial. But we are looking for a zero in between4 and 5. So, we will use the x^2 - 4x + 1 to find it.

We can use quadratic formula.

`x= (-(-4) +- sqrt((-4)^2-4(1)(-1)))/((2)(1))`

`x = (4 +- sqrt(20))/2`

Simplifying `sqrt(20).`

`x = (4 +- 2sqrt(5))/2`

Simplifying that will be.

`x = 2 +- sqrt(5)`

So, we will have:

`x = 2 + sqrt(5) =4.236067977` or `x = 2 - sqrt(5) =-0.2360679774`

Hence, the exact value of zero between 4 and 5 is **4.236067977**.