The polynomial P(x)=x3−4x2−5x−42 can be rewritten as P(x)=(x−6)(x2+2x+7). What can you conclude about P(x)?
There may be more than one correct answer. Select all correct answers.
When P(x) is divided by x−6, the remainder is x2+2x+7.
No factors can be identified from the rewritten expression.
When P(x) is divided by x2+2x+7, the remainder is x−6.
The binomial x−6 is a factor of P(x).
When P(x) is divided by x−6, the remainder is 0.
When P(x) is divided by x−6, the remainder is 0
no way
clearly,
The binomial x−6 is a factor of P(x).
what are the zeros
To determine which conclusions are correct about the polynomial P(x), we need to analyze the given polynomial and its rewritten expression.
The polynomial P(x) = x^3 - 4x^2 - 5x - 42 can be rewritten as P(x) = (x - 6)(x^2 + 2x + 7).
To verify the conclusions, we will apply polynomial division to determine the remainder when dividing P(x) by the given factors.
1. When P(x) is divided by x - 6, the remainder is x^2 + 2x + 7.
To verify this conclusion, perform polynomial long division with P(x) divided by x - 6. If the remainder is x^2 + 2x + 7, then the conclusion is correct.
2. No factors can be identified from the rewritten expression.
To verify this conclusion, analyze the rewritten expression (x - 6)(x^2 + 2x + 7). If it is not possible to factor this expression further, then the conclusion is correct.
3. When P(x) is divided by x^2 + 2x + 7, the remainder is x - 6.
To verify this conclusion, perform polynomial long division with P(x) divided by x^2 + 2x + 7. If the remainder is x - 6, then the conclusion is correct.
4. The binomial x - 6 is a factor of P(x).
To verify this conclusion, check if the factor x - 6 divides P(x) without leaving any remainder. If the division result is zero, then the conclusion is correct.
5. When P(x) is divided by x - 6, the remainder is 0.
To verify this conclusion, perform polynomial long division with P(x) divided by x - 6. If the remainder is zero, then the conclusion is correct.
Based on the analysis above:
- Conclusions 1, 3, and 4 are correct.
- Conclusion 2 is also correct since the rewritten expression cannot be factored further.
- Conclusion 5 is not correct because the remainder when dividing P(x) by x - 6 is x^2 + 2x + 7, not zero.
Therefore, the correct conclusions are:
- When P(x) is divided by x - 6, the remainder is x^2 + 2x + 7.
- No factors can be identified from the rewritten expression.
- When P(x) is divided by x^2 + 2x + 7, the remainder is x - 6.
- The binomial x - 6 is a factor of P(x).