To determine the remaining linear factors of the polynomial \( p(x) = x^4 + 7x^3 - x^2 - 7x \) given the known factors \( x \) and \( (x + 7) \), we can start by factoring \( p(x) \) using these known factors.
-
Since \( x \) is a factor, we can factor \( p(x) \) by \( x \): \[ p(x) = x(x^3 + 7x^2 - x - 7). \]
-
Next, we can factor the cubic polynomial \( x^3 + 7x^2 - x - 7 \) using the known factor \( (x + 7) \):
- To do this, we can use polynomial long division to divide \( x^3 + 7x^2 - x - 7 \) by \( (x + 7) \).
Performing the long division:
- Divide \( x^3 \) by \( x \) to get \( x^2 \).
- Multiply \( (x + 7) \) by \( x^2 \) to get \( x^3 + 7x^2 \).
- Subtract: \[ (x^3 + 7x^2 - x - 7) - (x^3 + 7x^2) = -x - 7. \]
- Divide \( -x \) by \( x \) to get \( -1 \).
- Multiply \( (x + 7) \) by \( -1 \) to get \( -x - 7 \).
- Subtract: \[ (-x - 7) - (-x - 7) = 0. \]
Thus, the division is exact, and we have: \[ x^3 + 7x^2 - x - 7 = (x + 7)(x^2 - 1). \]
-
Now we can rewrite the complete factorization of \( p(x) \): \[ p(x) = x(x + 7)(x^2 - 1). \]
-
The expression \( x^2 - 1 \) can be further factored as: \[ x^2 - 1 = (x - 1)(x + 1). \]
-
Therefore, the complete factorization of \( p(x) \) is: \[ p(x) = x(x + 7)(x - 1)(x + 1). \]
Final Answer: The remaining linear factors are \( (x - 1) \) and \( (x + 1) \).
1 answer