The polynomial equation for the given roots is x³-7x²+20x-24=0.
Given that, the polynomial equation of the smallest degree whose roots are 3, 2+2i and 2-2i.
Now, we need to write the polynomial equation.
<h3>What is a polynomial equation? </h3>
A polynomial equation is an equation where a polynomial is set equal to zero. i.e., it is an equation formed with variables, non-negative integer exponents, and coefficients together with operations and an equal sign. It has different exponents. The highest one gives the degree of the equation. For an equation to be a polynomial equation, the variable in it should have only non-negative integer exponents. i.e., the exponents of variables should be only non-negative and they should neither be negative nor be fractions.
Now, x=3, x=2+2i and x=2-2i
x-3, x-2-2i and x-2+2i
So, the polynomial equation is (x-3)(x-2-2i)(x-2+2i)=0
⇒(x-3)(x(x-2-2i)-2(x-2-2i)+2i(x-2-2i))=0
⇒(x-3)(x²-2x-2xi-2x+4+4i+2xi-4i-4i²)=0
⇒(x-3)(x²-2x-2x+4-4i²)=0
⇒(x-3)(x²-4x+4+4)=0 (∵i²=-1)
⇒(x-3)(x²-4x+8)=0
⇒x(x²-4x+8)-3(x²-4x+8)=0
⇒x³-4x²+8x-3x²+12x-24=0
⇒x³-7x²+20x-24=0
Therefore, the polynomial equation for the given roots is x³-7x²+20x-24=0.
To learn more about the polynomial equation visit:
brainly.com/question/20630027.
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