Answer:
the answer to the division problem is `1x + 6` (a polynomial).
Explanation:
To divide polynomials, we use the long division method. Here's how we can divide x² - 36 by x - 6:
x² - 36 |
--------
x - 6
First, we divide the leading coefficient of the numerator (which is 1) by the leading coefficient of the denominator (which is 1).
We get 1 / 1 = 1 as the first coefficient of our quotient.
x² - 36 |
--------
x - 6
1
Then, we multiply the quotient by the denominator and subtract it from the numerator.
We get (1 * (x - 6)) = x - 6, and x² - 36 - (x - 6) = x² - x - 30.
x² - 36 |
--------
x - 6
1
x - 6
----------
x² - x - 30
We repeat the process, dividing the leading coefficient of the new numerator (which is 1) by the leading coefficient of the denominator (which is 1).
We get 1 / 1 = 1 as the next coefficient of our quotient.
x² - 36 |
--------
x - 6
1
x - 6
----------
x² - x - 30
1
We multiply the quotient by the denominator and subtract it from the numerator.
We get (1 * (x - 6)) = x - 6, and x² - x - 30 - (x - 6) = x² - 2x + 24.
x² - 36 |
--------
x - 6
1
x - 6
----------
x² - x - 30
1
x - 6
---------------
x² - 2x + 24
We repeat the process one more time, dividing the leading coefficient of the new numerator (which is 1) by the leading coefficient of the denominator (which is 1).
We get 1 / 1 = 1 as the next coefficient of our quotient.
x² - 36 |
--------
x - 6
1
x - 6
----------
x² - x - 30
1
x - 6
---------------
x² - 2x + 24
1
We multiply the quotient by the denominator and subtract it from the numerator.
We get (1 * (x - 6)) = x - 6, and x² - 2x + 24 - (x - 6) = x² - 3x + 18.
x² - 36 |
--------
x - 6
1
x - 6
----------
x² - x - 30
1
x - 6
---------------
x² - 2x + 24
1
x - 6
------------------
x² - 3x + 18
Since the degree of the new numerator is less than the degree of the denominator, we have found the complete quotient.
So, the quotient is `1x + 6`, and the remainder is `x² - 3x + 18`.
Therefore, the answer to the division problem is `1x + 6` (a polynomial).