$1$ . Find all Roots of $P(x)=x^{4}-6 x^{3}-11 x^{2}+12 x-26$ Gmoi $3+2 i$ is a root

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Algebra 2

Quadratic equation with complex roots

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1

. Find all Roots of

P(x)=x^{4}-6 x^{3}-11 x^{2}+12 x-26

Gmoi

3+2 i

is a root

Identify Information: Identify the given information and the complex conjugate root theorem. $\newline$ Since $3 + 2i$ is a root of the polynomial $P(x)$ , by the complex conjugate root theorem, its conjugate $3 - 2i$ is also a root of $P(x)$ .
Find Quadratic Factor: Use the given roots to find a quadratic factor of $P(x)$ . We can write the factor associated with the roots $3 + 2i$ and $3 - 2i$ as $(x - (3 + 2i))(x - (3 - 2i))$ .
Expand Factor: Expand the quadratic factor. $\newline$ $(x - (3 + 2i))(x - (3 - 2i)) = (x - 3 - 2i)(x - 3 + 2i)$ $\newline$ $= x^2 - 3x + 2ix - 3x + 9 - 6i - 2ix + 6i - 4$ $\newline$ $= x^2 - 6x + 9 + 4$ $\newline$ $= x^2 - 6x + 13$
Divide Polynomial: Divide the polynomial $P(x)$ by the quadratic factor. $\newline$ We will perform polynomial long division or synthetic division to divide $P(x)$ by $x^2 - 6x + 13$ .
Perform Division: Perform the polynomial division. $\newline$ Since the division process is lengthy, we will not write out all the steps here. However, the result of dividing $P(x)$ by $x^2 - 6x + 13$ should be a quadratic polynomial.
Find Quotient Polynomial: Find the quotient polynomial. $\newline$ After dividing $P(x)$ by $x^2 - 6x + 13$ , we get a quotient polynomial of $x^2 - bx + c$ , where $b$ and $c$ are constants to be determined.
Use Product Relation: Use the fact that the product of the quadratic factor and the quotient polynomial equals $P(x)$ . $(x^2 - 6x + 13)(x^2 - bx + c) = x^4 - 6x^3 - 11x^2 + 12x - 26$
Expand Product: Expand the product and compare coefficients to find $b$ and $c$ . $\newline$ Expanding the product, we get: $\newline$ $x^4 - (6 + b)x^3 + (13 - 6b + c)x^2 - (6c + 13b)x + 13c$ $\newline$ Comparing coefficients with $P(x)$ , we find: $\newline$ $- (6 + b) = -6$ $\newline$ $13 - 6b + c = -11$ $\newline$ $- (6c + 13b) = 12$ $\newline$ $13c = -26$
Solve System of Equations: Solve the system of equations for $b$ and $c$ . From $13c = -26$ , we get $c = -2$ . Substituting $c = -2$ into the other equations, we can solve for $b$ . $- (6 + b) = -6$ implies $b = 0$ . $13 - 6b - 2 = -11$ implies $11 - 6b = -11$ , which confirms $b = 0$ .
Write Quotient Polynomial: Write down the quotient polynomial. $\newline$ The quotient polynomial is $x^2 - 0x - 2$ , which simplifies to $x^2 - 2$ .
Find Polynomial Roots: Find the roots of the quotient polynomial. $\newline$ The roots of $x^2 - 2$ are found by setting the polynomial equal to zero and solving for $x$ : $\newline$ $x^2 - 2 = 0$ $\newline$ $x^2 = 2$ $\newline$ $x = \pm\sqrt{2}$
List All Roots: List all the roots of $P(x)$ . The roots of $P(x)$ are $3 + 2i$ , $3 - 2i$ , $\sqrt{2}$ , and $-\sqrt{2}$ .