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I only need an example for this
these are pretty in-depth questions, but ill try to simplify the ideas. let's start with the last one by definition, a rational expression is any polynomial divided by any polynomial except for zero. For a rational expression to be "closed" under addition, mult, division, and subtraction, the following must apply: A rational expression plus a rational expression is a rational expression A rational expression minus a rational expression is a rational expression A rational expression times a rational expression is a rational expression A rational expression divided by a rational expression is a rational expression let us denote a rational expression as: a(x) / b(x) rational expressions are pretty much like fractions when it comes to adding/sub/mult/div so the following rules apply: ADDITION: a(x) / b(x) + c(x) / d(x) = (a(x) * d(x) + b(x) * c(x)) / (b(x) * d(x)) SUBTRACTION: a(x) / b(x) - c(x) / d(x) = (a(x) * d(x) - b(x) * c(x)) / (b(x) * d(x)) MULTIPLICATION: a(x) / b(x) * c(x) / d(x) = ( a(x) * c(x )) / ( b(x) * d(x) ) DIVISION: ( a(x) / b(x) ) / ( c(x) / d(x) ) = ( a(x) * d(x) ) / ( b(x) * c(x)) In each case we end up with a polynomial divided by a polynomial which satisfies our definition of a rational expression. Thus, rational expressions are closed under addition, subtraction, multiplication, and division. It might take you a while to soak that it, but it's not so bad.
I think I have time to help with the first one. As you probably know you can write a complex number in the for (a + bi) now, let's look at a polynomial identity. for example, if we have something x^3 - 64, we know that equals x^3 - 4^3 and the polynomial identity says that is equal to (x - 4)(x^2 +4x + 16) because we know that anything in the form (a^3 - b^3) = (a - b)(a^2 + ab + b^2) however, we know that we can use complex numbers to factor x^2 + 4x + 16 by using the quadratic formula after applying the quadratic formula, we get that x = -2 + 2isqrt(3) and x = -2 - 2isqrt(3)
if u need anything, just anything at all let me know
okay Thank you