# Pre-Calc Question?

Using the given zero, find one other zero of f(x). Explain the process you used to find your solution. (3 points)

1 - 6i is a zero of f(x) = x4 - 2x3 + 38x2 - 2x + 37.

• f(x) = x⁴ - 2x³ + 38x² - 2x + 37

If (1 - 6i) is a zero, it means that (1 + 6i) is a zero too.

So you can factorize: [x - (1 - 6i)].[x - (1 + 6i)]

= [x - (1 - 6i)].[x - (1 + 6i)]

= (x - 1 + 6i).(x - 1 - 6i)

= x² - x - 6xi - x + 1 + 6i + 6ix - 6i - 36i²

= x² - 2x + 1 - 36i² → where: i² = - 1

= x² - 2x + 37 → abd to obtain x⁴, it's necessary the first term of the other factor to be x²

= (x² - 2x + 37).(x² + ax + b) → you expand

= x⁴ + ax³ + bx² - 2x³ - 2ax² - 2bx + 37x² + 37ax + 37b → you group

= x⁴ + x³.(a - 2) + x².(b - 2a + 37) - x.(2b - 37a) + 37b → you compare with: x⁴ - 2x³ + 38x² - 2x + 37

37b = 37 → b = 1

(2b - 37a) = 2 → 37a = 2b - 2 → 37a = 0 → a = 0

(b - 2a + 37) = 38 → (1 - 2a + 37) = 38 → 38 - 2a = 38 → - 2a = 0 → a = 0 ← of course

(a - 2) = - 2 → a = 0 ← of course

Resart:

= (x² - 2x + 37).(x² + ax + b) → we've just seen that: a = 0

= (x² - 2x + 37).(x² + b) → recall:: b = 1

= (x² - 2x + 37).(x² + 1)

We've seen that (x² - 2x + 37) = 0 corespnds to: [x - (1 - 6i)].[x - (1 + 6i)] = 0

The other case is:

(x² + 1) = 0

x² + 1 = 0

x² = - 1

x² = i²

x = ± i

The other zero is (+ i) and (- i)

• You have a polynomial with real coefficients. Therefore any complex roots occur in conjugate pairs, (a+ib) and (a-ib). You are given one zero that is one member of a conjugate pair of roots.

It is trivial that another zero, such as you are asked to find, is the other member of that conjugate pair.

Ans: Given that x =1-6i is a zero of f(x), another zero is x = 1+6i

• If a+bi then a-bi is a root.

Lets call these r1 & r2.

Thus (x-r1) & (x-r2) are factors of f(x). Therefore, just factor these out of your f(x).

Or equivalently, expand (x-r1)*(x-r2) and factor out that resulting expression from f(x) by synthetic division, say.

You will be left with a quadratic, which you know how to find its roots (your last two roots!).

Show your steps here if need be and we can verify ur work.

Done!

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