Answer :
To divide the polynomial [tex]\( 3t^2 - 12t - 36 \)[/tex] by [tex]\( t - 6 \)[/tex], we can use polynomial long division. Below is a step-by-step breakdown of the process:
1. Set up the division:
[tex]\[
\frac{3t^2 - 12t - 36}{t - 6}
\][/tex]
2. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{3t^2}{t} = 3t
\][/tex]
So, the first term of the quotient is [tex]\( 3t \)[/tex].
3. Multiply [tex]\( 3t \)[/tex] by the divisor [tex]\( t - 6 \)[/tex]:
[tex]\[
3t \cdot (t - 6) = 3t^2 - 18t
\][/tex]
4. Subtract the result from the original polynomial:
[tex]\[
(3t^2 - 12t - 36) - (3t^2 - 18t) = -12t + 18t - 36 = 6t - 36
\][/tex]
5. Divide the new first term by the first term of the divisor:
[tex]\[
\frac{6t}{t} = 6
\][/tex]
So, the next term in the quotient is [tex]\( 6 \)[/tex].
6. Multiply [tex]\( 6 \)[/tex] by the divisor [tex]\( t - 6 \)[/tex]:
[tex]\[
6 \cdot (t - 6) = 6t - 36
\][/tex]
7. Subtract the result from the previous remainder:
[tex]\[
(6t - 36) - (6t - 36) = 0
\][/tex]
When the remainder is 0, this means the division is exact.
Putting it all together, our quotient is:
[tex]\[
3t + 6
\][/tex]
Therefore, the answer to the problem is:
[tex]\[
3(t + 2)
\][/tex]
1. Set up the division:
[tex]\[
\frac{3t^2 - 12t - 36}{t - 6}
\][/tex]
2. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{3t^2}{t} = 3t
\][/tex]
So, the first term of the quotient is [tex]\( 3t \)[/tex].
3. Multiply [tex]\( 3t \)[/tex] by the divisor [tex]\( t - 6 \)[/tex]:
[tex]\[
3t \cdot (t - 6) = 3t^2 - 18t
\][/tex]
4. Subtract the result from the original polynomial:
[tex]\[
(3t^2 - 12t - 36) - (3t^2 - 18t) = -12t + 18t - 36 = 6t - 36
\][/tex]
5. Divide the new first term by the first term of the divisor:
[tex]\[
\frac{6t}{t} = 6
\][/tex]
So, the next term in the quotient is [tex]\( 6 \)[/tex].
6. Multiply [tex]\( 6 \)[/tex] by the divisor [tex]\( t - 6 \)[/tex]:
[tex]\[
6 \cdot (t - 6) = 6t - 36
\][/tex]
7. Subtract the result from the previous remainder:
[tex]\[
(6t - 36) - (6t - 36) = 0
\][/tex]
When the remainder is 0, this means the division is exact.
Putting it all together, our quotient is:
[tex]\[
3t + 6
\][/tex]
Therefore, the answer to the problem is:
[tex]\[
3(t + 2)
\][/tex]
