Step-by-step rationalization: The expression represents a fancy quantity. Significantly, advanced options cannot occur in odd quantity, for instance, the minimal advanced options an expression can have is 2. Moreover, advanced options are associated by conjugates, which implies the second answer to this polynomial is , as a result of is the conjugate of the given advanced root.
One different zero is 2+3i Step-by-step rationalization: If 2-3i is a zero and all of the coefficients of the polynomial perform is actual, then the conjugate of 2-3i can be a zero. The conjugate of (a+b) is (a-b). The conjugate of (a-b) is (a+b). The conjugate of (2-3i) is (2+3i) so 2+3i can be a zero. Okay so we now have two zeros 2-3i and a couple of+3i. Which means that (x-(2-3i)) and (x-(2+3i)) are components of the given polynomial. I will discover the product of those components (x-(2-3i)) and (x-(2+3i)). (x-(2-3i))(x-(2+3i)) Foil! First: x(x)=x^2 Outer: x*-(2+3i)=-x(2+3i) Interior: -(2-3i)(x)=-x(2-3i) Final: (2-3i)(2+3i)=4-9i^2 (You’ll be able to simply do first and final when multiplying conjugates) ———————————Add collectively: x^2 + -x(2+3i) + -x(2-3i) + (4-9i^2) Simplifying: x^2-2x-3ix-2x+3ix+4+9 (since i^2=-1) x^2-4x+13 (since -3ix+3ix=0) So x^2-4x+13 is an element of the given polynomial. I will do lengthy division to seek out one other issue. Hopefully we get a the rest of 0 as a result of we’re saying it’s a issue of the given polynomial. x^2+1 ————————————— x^2-4x+13| x^4-4x^3+14x^2-4x+13 -( x^4-4x^3+ 13x^2) —————————————— x^2-4x+13 -(x^2-4x+13) —————– 0 So the opposite issue is x^2+1. To seek out the zeros of x^2+1, you set x^2+1 to 0 and clear up for x. So the zeros are i, -i , 2-3i , 2+3i
The opposite zero of f(x) is: 1+6i Step-by-step rationalization: We’re given a polynomial equation f(x) as: We all know that : For any polynomial perform with integer coefficients if there’s a advanced zero of that polynomial then the conjugate of that zero can be a zero. Right here we’re on condition that: 1-6i is a zero of the polynomial perform f(x) The advanced conjugate of 1-6i is: 1+6i Which means that: 1+6i is the opposite zero of the polynomial perform f(x).
So, the opposite options are
The Complicated conjugate root theorem states that if a-bi is an answer, so is a+bi, and vice versa. Due to this fact, 1+6i is one other answer
You want a graphing calculator which is like $150, however should you do not need to be broke simply use demos. com and put it in the way you see it. Hope this helps, oh and by the best way f(x) is identical as saying y.
Primarily based in your query ask to seek out one other zero off (x) 1-6i if 1-6i is root. since advanced roots at all times seem in pairs. the opposite root should be its conjugate. so the doable worth of the opposite zero is simply the identical root of the perform f(x) and that’s 1-6i
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