Question: Given that \(4 - 3i\) is a zero, factor the following polynomial function completely. Use the…

Given that \(4 - 3i\) is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

\[ f(x) = x^4 - 22x^3 + 182x^2 - 710x + 1125 \]

Answer:

Solution

Given that \(4 - 3i\) is a zero, we know its conjugate \(4 + 3i\) is also a zero. The polynomial is: \[ f(x) = x^4 - 22x^3 + 182x^2 - 710x + 1125 \] Since \(4 - 3i\) and \(4 + 3i\) are zeros, then \((x - (4 - 3i))(x - (4 + 3i)) = (x - 4 + 3i)(x - 4 - 3i)\) is a factor. This simplifies to a quadratic: \[ (x - 4)^2 - (3i)^2 \] Expanding this: \[ = x^2 - 8x + 16 - 9 \] \[ = x^2 - 8x + 25 \] Next, we divide the polynomial \(f(x)\) by this quadratic factor \(x^2 - 8x + 25\). Using polynomial long division, let’s perform the division step by step. 1. Divide the first term of the dividend \(x^4\) by the first term of the divisor \(x^2\) to get \(x^2\). 2. Multiply \(x^2\) by the entire divisor \(x^2 - 8x + 25\): \[ x^2(x^2 - 8x + 25) = x^4 - 8x^3 + 25x^2 \] 3. Subtract this from the original polynomial: \[ (x^4 - 22x^3 + 182x^2 - 710x + 1125) - (x^4 - 8x^3 + 25x^2) = -14x^3 + 157x^2 - 710x + 1125 \] 4. Repeat the process: divide \(-14x^3\) by \(x^2\) to get \(-14x\). 5. Multiply \(-14x\) by the divisor: \[ -14x(x^2 - 8x + 25) = -14x^3 + 112x^2 - 350x \] 6. Subtract: \[ (-14x^3 + 157x^2 - 710x + 1125) - (-14x^3 + 112x^2 - 350x) = 45x^2 - 360x + 1125 \] 7. Divide \(45x^2\) by \(x^2\) to get \(45\). 8. Multiply \(45\) by the divisor: \[ 45(x^2 - 8x + 25) = 45x^2 - 360x + 1125 \] 9. Subtract: \[ (45x^2 - 360x + 1125) - (45x^2 - 360x + 1125) = 0 \] So, the result of the division is \(x^2 - 14x + 45\). Thus, the factors of the polynomial are \((x^2 - 8x + 25)\) and \((x^2 - 14x + 45)\).