Question
What are the remaining linear factors of x4−13x3−x2+13x with known factors (x+1) and (x−1) ?(1 point)%0D%0AResponses%0D%0A%0D%0A(x2−13x)%0D%0Aleft parenthesis x squared minus 13 x right parenthesis%0D%0A%0D%0A(x−1) and (x−13)%0D%0Aleft parenthesis x minus 1 right parenthesis and left parenthesis x minus 13 right parenthesis%0D%0A%0D%0A(x2−13)%0D%0Aleft parenthesis x squared minus 13 right parenthesis%0D%0A%0D%0Ax and (x−13)
Answers
GPT-4o mini
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.
1. **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.
2. **Calculate the product of the known factors:**
\[ (x + 1)(x - 1) = x^2 - 1 \]
3. **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:
\[
P(x) = (x^2 - 1)(x^2 - 13)
\]
4. **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.
1. **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.
2. **Calculate the product of the known factors:**
\[ (x + 1)(x - 1) = x^2 - 1 \]
3. **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:
\[
P(x) = (x^2 - 1)(x^2 - 13)
\]
4. **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.