### Part a: Divide [tex]\( x^2 - x - 12 \)[/tex] by [tex]\( x - 4 \)[/tex]
To divide the polynomial [tex]\( x^2 - x - 12 \)[/tex] by [tex]\( x - 4 \)[/tex], we use polynomial long division:
1. Divide the leading term:
- Divide [tex]\( x^2 \)[/tex] by [tex]\( x \)[/tex], which gives [tex]\( x \)[/tex].
2. Multiply and subtract:
- Multiply [tex]\( x \)[/tex] by [tex]\( x - 4 \)[/tex] to get [tex]\( x^2 - 4x \)[/tex].
- Subtract [tex]\( x^2 - 4x \)[/tex] from [tex]\( x^2 - x - 12 \)[/tex]:
[tex]\[
(x^2 - x - 12) - (x^2 - 4x) = -x + 4x - 12 = 3x - 12
\][/tex]
3. Repeat the process:
- Divide [tex]\( 3x \)[/tex] by [tex]\( x \)[/tex], which gives [tex]\( 3 \)[/tex].
- Multiply [tex]\( 3 \)[/tex] by [tex]\( x - 4 \)[/tex] to get [tex]\( 3x - 12 \)[/tex].
- Subtract [tex]\( 3x - 12 \)[/tex] from [tex]\( 3x - 12 \)[/tex]:
[tex]\[
(3x - 12) - (3x - 12) = 0
\][/tex]
Hence, the quotient is [tex]\( x + 3 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
### Part b: Divide [tex]\( x^2 - 5x + 6 \)[/tex] by [tex]\( x - 2 \)[/tex]
To divide the polynomial [tex]\( x^2 - 5x + 6 \)[/tex] by [tex]\( x - 2 \)[/tex], we follow a similar polynomial long division process:
1. Divide the leading term:
- Divide [tex]\( x^2 \)[/tex] by [tex]\( x \)[/tex], which gives [tex]\( x \)[/tex].
2. Multiply and subtract:
- Multiply [tex]\( x \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( x^2 - 2x \)[/tex].
- Subtract [tex]\( x^2 - 2x \)[/tex] from [tex]\( x^2 - 5x + 6 \)[/tex]:
[tex]\[
(x^2 - 5x + 6) - (x^2 - 2x) = -5x + 2x + 6 = -3x + 6
\][/tex]
3. Repeat the process:
- Divide [tex]\( -3x \)[/tex] by [tex]\( x \)[/tex], which gives [tex]\( -3 \)[/tex].
- Multiply [tex]\( -3 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( -3x + 6 \)[/tex].
- Subtract [tex]\( -3x + 6 \)[/tex] from [tex]\( -3x + 6 \)[/tex]:
[tex]\[
(-3x + 6) - (-3x + 6) = 0
\][/tex]
Thus, the quotient is [tex]\( x - 3 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
### Summary
- Part a: The quotient of [tex]\( x^2 - x - 12 \)[/tex] divided by [tex]\( x - 4 \)[/tex] is [tex]\( x + 3 \)[/tex], with a remainder of [tex]\( 0 \)[/tex].
- Part b: The quotient of [tex]\( x^2 - 5x + 6 \)[/tex] divided by [tex]\( x - 2 \)[/tex] is [tex]\( x - 3 \)[/tex], with a remainder of [tex]\( 0 \)[/tex].
