# Without the use of the solve function of a calculator how do I solve the equation: #x^4-5x^3-x^2+11x-30=0#?

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Given that #(x-5)# and #(x+2)# are factors of #f(x)#

Given that

##### 2 Answers

The zeros are

#### Explanation:

We are told that

#x^4-5x^3-x^2+11x-30 = (x-5)(x^3-x+6)#

We are told that

#x^3-x+6 = (x+2)(x^2-2x+3)#

The discriminant of the remaining quadratic factor is negative, but we can still use the quadratic formula to find the Complex roots:

#x^2-2x+3# is in the form#ax^2+bx+c# with#a=1# ,#b=-2# and#c=3# .

The roots are given by the quadratic formula:

#x = (-b+-sqrt(b^2-4ac))/(2a)#

#= (2+-sqrt((-2)^2-(4*1*3)))/(2*1)#

#= (2+-sqrt(4-12))/2#

#= (2+-sqrt(-8))/2#

#= (2+-sqrt(8)i)/2#

#= (2+-2sqrt(2)i)/2#

#=1+-sqrt(2)i#

Let us try without knowing that

The constant term equals the roots product,so

This coefficient is an integer value whose factors are

We can represent the polynomial as

Calculating the right side and comparing both sides we obtain

Solving for

Evaluating the roots of