To find the other factor when x – 9 is a known factor of the polynomial x² + 5x + 36, we can use polynomial long division.
1. Divide the polynomial x² + 5x + 36 by x – 9.
2. The first term of the result will be x because when you multiply x – 9 by x, you get x² – 9x.
3. Subtract this from the original polynomial: (x² + 5x + 36) – (x² – 9x) = 14x + 36.
4. Now, divide 14x by the leading term x from x – 9, which gives you +14.
5. Multiply (x – 9) by 14 to get 14x – 126.
6. Subtract again: (14x + 36) – (14x – 126) = 162.
The result shows that after dividing, we have a remainder of 162, which means x – 9 is not actually a factor of the polynomial x² + 5x + 36. Thus, we need to evaluate the problem correctly, or x – 9 must be correctly factored with respect to a different polynomial.
In this particular polynomial, it signifies that either x – 9 is not a factor, or the quadratic expression does not divide evenly with this factor.