Pre-Calc Question?
Using the known zero (1 - 6i) of the function (f(x) = x^4 - 2x^3 + 38x^2 - 2x + 37), can you find another zero of (f(x))? Please explain the process you used to arrive at your solution. (3 points)
3 Answers
Feb 01, 2025
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
Jan 09, 2025
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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