Given 1+2i as a zero of f(x) = x^4 - 2x^3 - 4x^2 + 18x - 45, find all other zeros and write the complete factorization. Please show work if you can.

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Step-by-step explanation:

One of the zeros given:

Since it is a complex number, its conjugate is also a zero:

The product of two factors are:

We need two more zeros. Let them be m and n, then we have:

The product of all factors:

Compare left and right sides and solve for m and n:

From the above equations we get:

The other zero's are 3 and -3

All of the factors:

(x−3)(x+3)(x^2-2x+5)

Step-by-step explanation:

f(x) = x^4 - 2x^3 - 4x^2 + 18x - 45

To factor the expression, solve the equation where it equals to 0.

x^4 - 2x^3 - 4x^2 + 18x - 45 = 0

By Rational Root Theorem, all rational roots of a polynomial are in the form p/q where p divides the constant term −45 and q divides the leading coefficient 1. List all candidates p/q.

±45,±15,±9,±5,±3,±1

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

x=3

By Factor theorem, x−k is a factor of the polynomial for each root k. Divide x^4 - 2x^3 - 4x^2 + 18x - 45 by x−3 to get x^3 + x^2 - x + 15. To factor the result, solve the equation where it equals to 0.

x^3 + x^2 - x + 15 = 0

By Rational Root Theorem, all rational roots of a polynomial are in the form p/q where p divides the constant term 15 and q divides the leading coefficient 1. List all candidates p/q.

±15,±5,±3,±1

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

x=−3

By Factor theorem, x−k is a factor of the polynomial for each root k. Divide x^3 + x^2 - x + 15 by x+3 to get x^2 - 2x + 5. To factor the result, solve the equation where it equals to 0.

x^2−2x+5=0

All equations of the form ax^2 + bx + c=0 can be solved using the quadratic formula: −b±√b2-4ac/2a. Substitute 1 for a, −2 for b, and 5 for c in the quadratic formula.

x=-(-2)+√(-2)^2 - 4 x 1 x 5 / 2

Do the calculations.

x=2+√-16 / 2

Polynomial x^2−2x+5 is not factored since it does not have any rational roots.

x^2−2x+5

Rewrite the factored expression using the obtained roots.

(x−3)(x+3)(x^2-2x+5)

thats not a system of equations, that is a solution

1

x = 1

step-by-step explanation:

0 ÷ x = 0

0 ÷ 1 = 0