Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

-73+98 i is a root of 
f(x)=x^(2)+146 x+ 14933. Find the other roots of 
f(x).
Write your answer as a list of simplified values separated by commas, if there is more than one value.

73+98i -73+98 i is a root of f(x)=x2+146x+ f(x)=x^{2}+146 x+ 1493314933. Find the other roots of f(x) \mathrm{f}(x) .\newlineWrite your answer as a list of simplified values separated by commas, if there is more than one value.

Full solution

Q. 73+98i -73+98 i is a root of f(x)=x2+146x+ f(x)=x^{2}+146 x+ 1493314933. Find the other roots of f(x) \mathrm{f}(x) .\newlineWrite your answer as a list of simplified values separated by commas, if there is more than one value.
  1. Complex Roots Conjugate Pairs: Since the coefficients of the polynomial are real numbers, the complex roots of polynomials with real coefficients come in conjugate pairs. This means that if 73+98i-73 + 98i is a root, then its conjugate, 7398i-73 - 98i, must also be a root.
  2. Sum of Roots Quadratic Equation: To find the other root, we can use the fact that the sum of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is equal to ba-\frac{b}{a}. In this case, the sum of the roots is 146-146.
  3. Setting up Equation for Other Root: We already know one of the roots is 73+98i-73 + 98i, so we can set up the equation 146=(73+98i)+other_root-146 = (-73 + 98i) + \text{other\_root} to solve for the other root.
  4. Calculating Other Root: Subtracting the known root from 146-146 gives us the other root: other_root=146(73+98i)=146+7398i=7398i\text{other\_root} = -146 - (-73 + 98i) = -146 + 73 - 98i = -73 - 98i.
  5. Final Result: Therefore, the other root of the polynomial f(x)f(x) is 7398i-73 - 98i.

More problems from Complex conjugate theorem