To divide the polynomial [tex]\(12x^3 - 6x^2 - 3x\)[/tex] by [tex]\(-3x\)[/tex], follow these steps carefully:
1. Set up the division: We start with the polynomial [tex]\(12x^3 - 6x^2 - 3x\)[/tex] and we are dividing it by [tex]\(-3x\)[/tex].
2. Divide the first term:
[tex]\[
\frac{12x^3}{-3x} = -4x^2
\][/tex]
Write [tex]\(-4x^2\)[/tex] as the first term of the quotient.
3. Multiply and subtract:
Multiply [tex]\(-4x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
-4x^2 \cdot (-3x) = 12x^3
\][/tex]
Subtract [tex]\(12x^3\)[/tex] from [tex]\(12x^3 - 6x^2 - 3x\)[/tex]:
[tex]\[
(12x^3 - 6x^2 - 3x) - 12x^3 = -6x^2 - 3x
\][/tex]
4. Divide the next term:
[tex]\[
\frac{-6x^2}{-3x} = 2x
\][/tex]
Write [tex]\(2x\)[/tex] as the next term of the quotient.
5. Multiply and subtract:
Multiply [tex]\(2x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
2x \cdot (-3x) = -6x^2
\][/tex]
Subtract [tex]\(-6x^2\)[/tex] from [tex]\(-6x^2 - 3x\)[/tex]:
[tex]\[
(-6x^2 - 3x) - (-6x^2) = -3x
\][/tex]
6. Divide the next term:
[tex]\[
\frac{-3x}{-3x} = 1
\][/tex]
Write [tex]\(1\)[/tex] as the next term of the quotient.
7. Multiply and subtract:
Multiply [tex]\(1\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
1 \cdot (-3x) = -3x
\][/tex]
Subtract [tex]\(-3x\)[/tex] from [tex]\(-3x\)[/tex]:
[tex]\[
-3x - (-3x) = 0
\][/tex]
Since we have no remainder left after the final subtraction, the quotient of the division is:
[tex]\[
-4x^2 + 2x + 1
\][/tex]
So, the quotient is [tex]\(-4x^2 + 2x + 1\)[/tex] and the remainder is [tex]\(0\)[/tex].
