To find the remaining linear factors of the polynomial \( x^4 - 13x^3 - x^2 + 13x \) given the factors \( (x + 1) \) and \( (x - 1) \), we can start by performing polynomial long division or synthetic division.
Step 1: Checking the known factors
Let's check if \( (x + 1) \) and \( (x - 1) \) are indeed factors by substituting \( x = -1 \) and \( x = 1 \) into the polynomial.
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For \( x = -1 \): \[ (-1)^4 - 13(-1)^3 - (-1)^2 + 13(-1) = 1 + 13 - 1 - 13 = 0 \] So \( x + 1 \) is a factor.
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For \( x = 1 \): \[ (1)^4 - 13(1)^3 - (1)^2 + 13(1) = 1 - 13 - 1 + 13 = 0 \] So \( x - 1 \) is also a factor.
Step 2: Polynomial division
We'll now divide \( x^4 - 13x^3 - x^2 + 13x \) by \( (x + 1)(x - 1) = x^2 - 1 \).
Perform the division:
- Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).
- Multiply \( x^2 \) by \( x^2 - 1 \) to get \( x^4 - x^2 \).
- Subtract: \[ (x^4 - 13x^3 - x^2 + 13x) - (x^4 - x^2) = -13x^3 + 13x \]
- Now divide \(-13x^3\) by \(x^2\) to get \(-13x\).
- Multiply \(-13x\) by \(x^2 - 1\) to get \(-13x^3 + 13x\).
- Subtract: \[ (-13x^3 + 13x) - (-13x^3 + 13x) = 0 \]
The polynomial factors as follows: \[ x^4 - 13x^3 - x^2 + 13x = (x^2 - 1)(x^2 - 13x) \]
Step 3: Factoring further
Now we need to factor \( x^2 - 13x \): \[ x^2 - 13x = x(x - 13) \]
Step 4: Complete Factorization
Putting it all together, we have: \[ x^4 - 13x^3 - x^2 + 13x = (x + 1)(x - 1)(x)(x - 13) \]
Final Answer
The remaining linear factors are:
- \( x \)
- \( (x - 13) \)
So the remaining linear factors are \( x \) and \( (x - 13) \).
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