To find the remaining linear factors of the polynomial \( x^4 + 7x^3 - x^2 - 7x \) given the known factors \( x \) and \( (x + 7) \), we start by factoring the polynomial.
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Start with the given polynomial:
\[ x^4 + 7x^3 - x^2 - 7x \] -
Factor out the known factors \( x \) and \( (x + 7) \):
Since one factor is \( x \), we can factor \( x \) out:
\[ x(x^3 + 7x^2 - x - 7) \] -
Next, we will factor \( x^3 + 7x^2 - x - 7 \). Since we know another factor is \( (x + 7) \), we can perform polynomial long division or synthetic division to divide \( x^3 + 7x^2 - x - 7 \) by \( (x + 7) \).
- Using synthetic division:
\[ \begin{array}{r|rrrr} -7 & 1 & 7 & -1 & -7 \ & & -7 & 0 & 7 \ \hline & 1 & 0 & -1 & 0 \ \end{array} \] The result of this division is: \[ x^2 - 1 \]
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Thus, we can write the polynomial as: \[ x(x + 7)(x^2 - 1) \]
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Next, we factor \( x^2 - 1 \) further: \[ x^2 - 1 = (x - 1)(x + 1) \]
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Putting all the factors together: \[ x(x + 7)(x - 1)(x + 1) \]
Now, the linear factors of the polynomial \( x^4 + 7x^3 - x^2 - 7x \) are \( x \), \( (x + 7) \), \( (x - 1) \), and \( (x + 1) \).
- Therefore, the remaining linear factors are \( (x + 1) \) and \( (x - 1) \).
The correct response is: (x + 1) and (x - 1).
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