Given that $1+2i$ is a zero of $k(x)=x^{4}-6x^{3}+26x^{2}-46x+65$ , find the remaining zeroes

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Algebra 2

Evaluate expression when a complex numbers and a variable term is given

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Q. Given that

1+2i

is a zero of

k(x)=x^{4}-6x^{3}+26x^{2}-46x+65

, find the remaining zeroes

Given Zeroes: Given that $1+2i$ is a zero of the polynomial $k(x)$ , we know that its complex conjugate $1-2i$ is also a zero of the polynomial because the coefficients of the polynomial are real numbers.
Factorization: Now we can write down the factors of $k(x)$ that correspond to these two zeroes: $(x - (1+2i))(x - (1-2i))$ We can expand this to find a quadratic factor of $k(x)$ .
Quadratic Factor: Expanding the factors we get: $\newline$ $(x - 1 - 2i)(x - 1 + 2i)$ $\newline$ = $((x - 1) - 2i)((x - 1) + 2i)$ $\newline$ = $(x - 1)^2 - (2i)^2$ $\newline$ = $x^2 - 2x + 1 - (-4)$ $\newline$ = $x^2 - 2x + 5$ $\newline$ This is a quadratic factor of $k(x)$ .
Polynomial Division: Since $k(x)$ is a fourth-degree polynomial and we have found a quadratic factor, we can perform polynomial division to find the other quadratic factor. We divide $k(x)$ by the quadratic factor we found: $\frac{k(x)}{x^2 - 2x + 5}$
Remaining Quadratic Factor: Performing the polynomial division, we get: $\newline$ $(x^4 - 6x^3 + 26x^2 - 46x + 65) \div (x^2 - 2x + 5)$ $\newline$ This should give us another quadratic polynomial, which will have the remaining two zeroes.
Finding Zeroes: The polynomial division yields: $\newline$ Quotient: $x^2 - 4x + 13$ $\newline$ Remainder: $0$ $\newline$ This means that the other quadratic factor of $k(x)$ is $x^2 - 4x + 13$ .
Calculating Discriminant: Now we need to find the zeroes of the quadratic polynomial $x^2 - 4x + 13$ . We can use the quadratic formula to find these zeroes: $\newline$ x = rac{-(-4) \[5pt] \pm \sqrt{(-4)^2 - 4(1)(13)}}{2(1)}
Applying Quadratic Formula: Calculating the discriminant: $\newline$ $(-4)^2 - 4(1)(13) = 16 - 52 = -36$ $\newline$ Since the discriminant is negative, the zeroes will be complex numbers.
Applying Quadratic Formula: Calculating the discriminant: $\newline$ $(-4)^2 - 4(1)(13) = 16 - 52 = -36$ $\newline$ Since the discriminant is negative, the zeroes will be complex numbers.Applying the quadratic formula: $\newline$ $x = \frac{4 \pm \sqrt{-36}}{2}$ $\newline$ $x = \frac{4 \pm 6i}{2}$ $\newline$ $x = 2 \pm 3i$ $\newline$ These are the remaining two zeroes of $k(x)$ .