To find the remaining linear factors of the polynomial \( x^4 - 13x^3 - x^2 + 13x \) given the known factors \( (x + 1) \) and \( (x - 1) \), we can start by performing polynomial long division or synthetic division to divide the given polynomial by the known factors.
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Factor out the known factors: Since \( (x + 1) \) and \( (x - 1) \) are factors, we can represent the polynomial as: \[ P(x) = (x + 1)(x - 1)(Q(x)) \] where \( Q(x) \) is a quadratic polynomial.
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Calculate the product of the known factors: \[ (x + 1)(x - 1) = x^2 - 1 \]
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Divide the polynomial by \( x^2 - 1 \): We perform polynomial division: \[ P(x) = x^4 - 13x^3 - x^2 + 13x \] Dividing by \( x^2 - 1 \):
- The first term is \( x^4 \div x^2 = x^2 \).
- Multiply \( x^2 \) by \( x^2 - 1 \) gives \( x^4 - x^2 \). Subtract this from \( P(x) \): \[ (x^4 - 13x^3 - x^2 + 13x) - (x^4 - x^2) = -13x^3 + 13x \]
- The next term is \( -13x^3 \div x^2 = -13x \).
- Multiply \( -13x \) by \( x^2 - 1 \) gives \( -13x^3 + 13x \). Subtract: \[ (-13x^3 + 13x) - (-13x^3 + 13x) = 0 \]
Since the remainder is 0, we conclude that: \[ P(x) = (x^2 - 1)(x^2 - 13) \]
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Factor \( x^2 - 13 \): The quadratic \( x^2 - 13 \) can be factored as: \[ x^2 - 13 = (x - \sqrt{13})(x + \sqrt{13}) \]
Thus, the linear factors of \( x^4 - 13x^3 - x^2 + 13x \) which remain are:
- \( (x - \sqrt{13}) \)
- \( (x + \sqrt{13}) \)
Therefore, the complete factorization of the polynomial is: \[ (x + 1)(x - 1)(x - \sqrt{13})(x + \sqrt{13}) \]
Among the response choices provided, the correct remaining linear factors based on the problem's clue would be:
- (x - sqrt(13)) and (x + sqrt(13)).
These factors represent the remaining roots of the polynomial.
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