Questions: Use synthetic division and the Factor Theorem to determine whether the given binomial is a factor of P(x). P(x)=25x^4-20x^3+13x^2+23x+5, x-1/5

Transcript text: Use synthetic division and the Factor Theorem to determine whether the given binomial is a factor of $P(x)$. \[ P(x)=25 x^{4}-20 x^{3}+13 x^{2}+23 x+5, \quad x-\frac{1}{5} \]

Solution

Solution Steps

To determine whether $ x - \frac{1}{5} $ is a factor of $ P(x) $ using synthetic division, we need to evaluate $ P \left( \frac{1}{5} \right) $. If the remainder is zero, then $ x - \frac{1}{5} $ is a factor of $ P(x) $; otherwise, it is not.

Step 1: Evaluate $ P\left(\frac{1}{5}\right) $

We perform synthetic division of $ P(x) = 25x^4 - 20x^3 + 13x^2 + 23x + 5 $ by $ x - \frac{1}{5} $. The coefficients of $ P(x) $ are $ [25, -20, 13, 23, 5] $ and the root to test is $ \frac{1}{5} = 0.2 $.

Step 2: Calculate the Remainder

After performing synthetic division, we obtain the result $ [25, -15.0, 10.0, 25.0, 10.0] $. The last element of this result, which represents the remainder, is $ 10.0 $.

Step 3: Determine if $ x - \frac{1}{5} $ is a Factor

Since the remainder $ 10.0 $ is not equal to $ 0 $, we conclude that $ x - \frac{1}{5} $ is not a factor of $ P(x) $.