# How do you construct polynomial equations with the given roots?

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**1.** #2# , #4# and #-7# .

**2.** #5# and #sqrt(3)# .

**1.**

**2.**

##### 1 Answer

**1.**

**2.**

#### Explanation:

Note that if a polynomial in

For question **1** we can construct a polynomial:

#f(x) = (x-2)(x-4)(x+7) = x^3+x^2-34x+56#

Any polynomial with these zeros will be a multiple (scalar or polynomial) of this

So the polynomial equation:

#x^3+x^2-34x+56 = 0#

has roots

For question **2** we can multiply out

#(x-5)(x-sqrt(3)) = x^2-(5+sqrt(3))x+5sqrt(3)#

If - as is probably the case - we want a polynomial with integer coefficients, then we also need the rational conjugate

Then we can define:

#g(x) = (x-5)(x-sqrt(3))(x+sqrt(3)) = (x-5)(x^2-3) = x^3-5x^2-3x+15#

Any polynomial with these zeros will be a multiple (scalar or polynomial) of this

So the polynomial equation:

#x^3-5x^2-3x+15 = 0#

has roots