The nth-degree polynomial function f(x) = x³ -4x²+ 22x +68 with n as zero of the polynomial.
<h3>
What is Zero of the Polynomial?</h3>
The polynomial's zeros are the locations at which its overall value is zero. In plain English, we might say that the zeros of a polynomial are the values of the variable at which the polynomial equals 0.
The roots of the equation are another name for a polynomial's zeros, which are frequently written as α and β. Grouping, factorization, and the usage of algebraic expressions are a few techniques for locating polynomial zeros.
GIVEN
-2 is a zero of the polynomial.
3 + 5i is a zero of the polynomial.
Therefore,
3 + 5i has conjugate 3 - 5i is also a zero.
Therefore,
(x +2) , (x-(3+5i)) and (x -(3-5i)) are factors of f(x).
Therefore,
f(x) = a(x + 2) (x-(3+5i)) (x -(3-5i))
Now solving it further we get
f(x) = a(x + 2) (x-3) + 5i) (x -3) -5i)
f(x) = a(x + 2) {(x-3)² - 25i²}
f(x) = a(x + 2) {(x-3)² + 25} ( since i² = -1)
Also, f(-1) = 41 and x = -1
then 41 = a(-1 + 2) {(-1-3)² + 25}
41 = a ×1 × 41
a = 1
Lastly
f(x) = (x + 2) {(x-3)² +25}
f(x) = (x + 2) {x²+ 9 - 6x + 25}
f(x) = (x + 2) {x² - 6x + 34}
f(x) = x³ -4x²+ 22x +68
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